MAT 003 Brian Killough s Instructor Notes Saint Leo University

MAT 003 Brian Killough s Instructor Notes Saint Leo University Success in online courses requires self-motivation and discipline. It is anticipated that students will read the textbook and complete sample exercises as required, before working on the My-Math-Lab (MML) modules. In general, our textbook is a reference and all of our homework and tests are completed within the MML modules. A summary of the modules and corresponding textbook material is in the table below. These instructor notes are intended to supplement the course by providing a short summary of the key material and providing solutions to example problems that emphasizing important or confusing topics. I hope you find them useful. Module Textbook Material 1 Sections 1.1 to 1.8 2 Sections 2.1 to 2.4 3 Sections 2.5 to 2.6 4 Sections 4.1 to 4.3 5 Sections 4.4 to 4.6 6 Sections 5.1 to 5.4 7 Sections 6.1, 6.2, 6.4, and 6.7 8 Sections 5.5 and 10.6 Chapter 1 Variables, Real Numbers and Mathematical Models MML Module #1 Section 1.1 Introduction to Algebra: Variables and Mathematical Models It is possible to convert standard English expressions into Algebraic Expressions using variables (such as X,Y,Z) and numbers. For example, The sum of a number and 6... x + 6 Five times a number decreased by 3... 5x - 3 Evaluating Expressions requires that we substitute numbers into variable expressions. For example, if we want to evaluate the algebra expression 5(x+3) for x = 5, then we replace 5 for the variable "x". 5(x+3)... 5(5+3) = 5(8) = 5*8 = 40 Section 1.2 Fractions in Algebra Converting improper fractions to mixed numbers (and the reverse) is an essential process. An improper fraction is just a single fraction (has a top and bottom part) that can be converted to a whole number and some leftover parts. Here are a few examples: : Convert to an improper fraction. First, take the denominator (bottom) and multiply it times the whole number on the left (2). This gives us 18. Now add that to the top (7) to get 25 on the top. The answer is then. Notice, how we keep the same denominator on the bottom, but the top is now

larger. You can think of the original 2 out front as really the same as. When we add that to the right side (7/9) we get the end result of 25/9. Now lets talk about simplifying fractions this is very important. We ALWAYS want to reduce fractions into their simplest form by simplifying them. : Simplify we can also write this as 18/45, when typing on one line. Now, to reduce we must find the Greatest Common Factor (GCF) of the top and bottom. This is the largest common number that is part of BOTH the top and bottom. To find this, it is best to take the fraction apart separate it, into it parts to find the factors that can be reduced. To do this, break down the fraction into the smallest multiples you can. now, we can cancel out terms that are common on the top and bottom. So in this case, we can remove a pair of 3 s from the top and bottom. That leaves us with 2/5 as the answer. Multiplying Fractions: When we multiply fractions, we can multiply the tops and then the bottoms, separately. Also, remember a whole number (like 9) can ALWAYS be written as a fraction, such as 9/1. So lets try this one : Simplify and Reduce Multiply the TOP terms to get 2. Then multiply the bottom terms to get 24. Now, simplify as we did in the previous example. So find the factors on the top and bottom. When we eliminate the 2 on top, it leaves a 1 behind this is important. It DOES NOT leave a zero behind, but always a 1. Dividing Fractions: When we divide fractions, we actually MULTIPLY. We take the division sign and change it to a multiplication. Then we must FLIP-OVER the second part to create what is called the RECIPROCAL of the second term. Then we multiply, like above. Lets try one Simplify and Reduce Notice the division is converted to a multiplication. Then the second term is FLIPPED over. Then we multiply and simplify. When dividing fractions we convert the division into a multiplication and flip over (called reciprocal) the following fraction. This is called INVERT and MULTIPLY Adding and Subtracting in order to add or subtract fractions, they MUST have the same denominators (bottom terms). To make this work, we have to be able to convert fractions. Simplify and Reduce 3 2 2 3 notice the bottom numbers are different. We cannot perform this subtraction yet. The bottom terms MUST be the same. So, first we have to find the Least Common Denominator (LCD) and then convert each fraction. The LCD is the smallest common number that the denominators from each fraction can be divided into. One way to find this, is to multiply both bottom terms together so we get 2*3 = 6. This could be used as an LCD. So lets convert each fraction into a denominator of 6.

First Term: 3/2 to get a 6 on the bottom, we must multiply both the top and bottom by 3. We are allowed to do this, since multiplying by 3/3 is the same as multiplying by 1, and that does not change the value of the fraction. So we get: Second Term: 2/3 to get a 6 on the bottom, we must multiply both the top and bottom by 2. So we get: Now that the bottoms are the same, we can perform the subtraction: When we add or subtract fractions, we only perform the operation on the top part of the fraction. The bottom denominator stays the same. Be sure to practice these types of problems.. there are many in the book and they are very important to master!!! Section 1.3 The Real Numbers It is important to know the various types of mathematical numbers. This is best done by describing the categories of numbers and giving some examples. Name Description s Natural Numbers Counting Numbers 1,2,3,4,5 Whole Numbers Counting Numbers + ZERO 0,1,2,3,4,5 Integers Number Line (negatives + positives) -3,-2,-1,0,1,2,3 Rational Numbers Can be written as a fraction, 2=2/1, 2/3=0.66666, (includes integers and repeating decimals) Irrational Numbers Square roots that cannot be reduced, PI, Natural Number (e) Non-Real Dividing by Zero, Square roots of negative numbers these are known as UNDEFINED numbers. If you are comparing the value of two numbers (with inequality signs), then it is best to convert a number into a decimal. Absolute Value: Absolute value is merely the POSITIVE equivalent of any number: Expressing Rational Numbers (fractions) as Decimals: The easiest way to convert a fraction to a decimal is to perform the division on your calculator. For example, if you are given 3/4, then the division on a calculator will give us: 0.75. Section 1.4 Basic Rules of Algebra Commutative Property switch the order of operations (a + b = b + a) Associative Property switch parenthesis 2 + (3+1) = (2+3) + 1 Distributive Property distribute a multiplier inside the parenthesis 2(a+b) = 2a + 2b : 4 (x+6) = 4x + 24 The properties can be used to strategically solve problems by combining terms and arranging operations to make things easier.

Like terms with variables should be combined: : 7x + 8 + 2x 3 = 9x + 5 Multiplication does not distribute over parenthesis unless there is addition. : Section 1.5 8 (5x) = 40x The wrong answer is 8(5) + 8x = 40 + 8x WRONG Addition of Real Numbers Identity Property of Addition achieve an identical term : a+0=a Inverse Property of Addition uses the inverse term to simplify : a + (-a) = 0 The book teaches this section using the idea of a NUMBER line. I prefer to simply use common addition and subtraction, which is similar to how it is done on the calculator. One trick to remember, is that we can always convert the addition of a NEGATIVE term, into a subtraction. For example: 3 + (-2) = 3 2 = 1 So lets try some example problems from the book -4 + (-6) = -4-6 = -10 10 + (-3) + (-8) = 10 3 8 = 10-11 = -1 Notice how the second term had to be converted to the same denominator as the first term. This is described in Section 1.2 (above). We found the LCD=10 and then multiplied the second term by 2/2 to get a 10 on the bottom. Then we can perform the subtraction. Section 1.6 8(3-7a) + 4(2a+3) = 24-56a+8a+12 = 36 48a Subtraction of Real Numbers Addition and subtraction is most difficult when there are different signs. We can always think of subtraction as just adding a negative. Also, remember that subtracting a negative is the same as adding since the two negative signs will cancel each other. : A B = A + ( - B) : A ( - B) = A + B 5 ( -17) = 5 + 17 = 22-6 (-3) + 8-11 = -6 + 3 + 8-11 = -3+8-11 = 5-11 = -6 NOTE: Some calculators (TI-83) have a - sign for arithmetic (subtraction) and a (-) sign for making a number negative. BE CAREFUL when entering negative numbers on your calculator. Section 1.7 Multiplication and Division of Real Numbers Identity Property of Multiplication achieve an identical term : Inverse Property of Multiplication uses the inverse term to simplify :

Multiplication Property of Zero always achieve zero when multiplying by zero : Remember the product of two negative numbers is a positive number. Think of the signs canceling. (-11)(-1) = 11 Some books use an X or a DOT for multiplication symbols. If there is no symbol, it is implied that we multiply. (10)(-7) = -70 Think of division as merely multiplying by a reciprocal. The reciprocal is the number in fraction form flipped over. Given 2. the reciprocal is: Given.. the reciprocal is: 3 Your book refers to this as the Multiplicative Inverse. If you take the original number and multiply it by its multiplicative inverse, you will ALWAYS get 1. So lets try some division problems NEVER divide by ZERO that is undefined. This one is OK since we are dividing by -10. Notice how the 8 was converted to fraction 8/1. Also, the division problem was converted to a multiplication by flipping the second term (multiplicative inverse). At the end, we simplify and reduce. Section 1.8 Exponents, Order of Operations and Mathematical Models This is a VERY important section. If you can do these, you are in VERY good shape for further algebra problems. Exponential Expressions: These are often called powers and represent a simplified method to write many multiplications. where x is the base and 5 is the power. Note: DO NOT multiply the base and the power but think of expanding the exponent.

Order of Operations: You may remember the saying Please Excuse My Dear Aunt Sally which uses the first letter of each word to represent the order of operations.. that is Parenthesis Exponents Multiplication and Division Addition and Subtraction When simplifying expressions we must use these rules in the proper order: (1) Work from the inside out, if there are multiple sets of parenthesis (2) Evaluate parenthesis also consider absolute value symbols as well (3) Evaluate exponents (left to right) (4) Perform multiplication and division (left to right) (5) Perform addition and subtractions (left to right) NOTE: Always go left to right if you have several of the same type of operations (multiplication and division, or addition and subtraction). Now lets try a few examples follow them closely This is a tough problem. We must use a common denominator on the top to simplify those terms. Then, we convert the large division problem into a multiplication, by flipping the second term over reciprocal. In the end, we simplify. See if you can follow the details. Chapter 2 Linear Equations and Inequalities in One Variable MML Module #2 Section 2.1 The Addition Property of Equality This section focuses on solving an algebra equation in ONE variable. IMPORTANT: The rule to remember in this chapter is whatever mathematical process you do on one side of an equation (add, subtract, multiply, divide), you must perform the same process on the other side of the equation. You may add or subtract numbers from both sides of an equation to get a variable by itself. This is called isolating the variable and is essential to solving equations. By isolation we mean, getting the variable on to one side of the equation and the numbers on the opposite side.

Any solution to an equation can be easily verified using substitution. You should get into a habit of taking the time to check your solutions. Replace your answer back into the ORIGINAL equation. If you use an equation from the middle of your work it may contain an error. When working with equations you can move a term across the equal sign, but you must change its sign. In the previous problem if you moved the 7 to other side it would become 7 and then could be combined with 19 to get 26. move the 3/5 to the other side to get -3/5. notice how 3/5 was converted to 6/10 to perform the subtraction using common denominators. Section 2.2 The Multiplication Property of Equality When solving equations, we are often left with a variable that has a number in front. To get rid of that number and solve for the variable, we must divide BOTH sides of the equation by that number. For example: -4y = 32 to solve for Y, we must divide BOTH sides by -4 to get: Y = -8 Just remember that dividing by 4 is the same as multiplying by 1/4. This is often easier to use. Finally, we may multiply by (-1) to get a sign change in a problem. All of these rules are needed to solve for variables. Section 2.3-3y 2 = -5 4y -3y + 4y = -5 + 2 y = -3 Solving Linear Equations If there are variables in several places we must follow these rules: (a) Add and subtract to move all the variables to one side. Recall if you move any term across an equal sign in an equation, it sign will change. This is the same as adding or subtracting. (b) Combine the variables and numbers. (c) Multiply and/or divide to isolate and solve for the variable.

Solving fractional equations can be difficult, but the best trick is to find the Least Common Denominator (LCD) and multiply EVERY term in the equation by the LCD to remove all fractions in one step. Note: Only consider the fractions when finding the Least Common Denominator (LCD). Whole numbers are really fractions with a ONE on the bottom and do not affect the selection of the LCD. Here are some examples There are 3 types of solutions you can find: 1. Conditional you get a standard solution, such as x=7, etc. 2. All Reals IDENTITY solution you end up with X=X, or 10=10, etc. 3. Inconsistent.. NO SOLUTION, Empty Set, Null Set you end up with NO VARIABLES, or 3=4 Now lets try some more examples from the book 3(3x-1)=4(3+3x) 9x-3 = 12+12x 9x-12x=12+3 then -3x=15, divide by -3 to get X = -5 This is a CONDITIONAL solution Any equation where the variable is eliminated and the numbers do not match is called INCONSISTENT or NO SOLUTION. 2(x+1)=2(x+3) then 2x+2=2x+6 then 2=6 (No solution or INCONSISTENT) Any equation where you end up with 0=0 in the end is called the IDENTITY solution or ALL REALS.

Section 2.4 Formula and Percents Solving a formula for one variable is an important skill. Here are some examples and tips. Also, be sure to check out examples 1 through 4 in the book. If the equation contains the variable in one place then isolate the variable (get it by itself) and solve like you did in section 2.3. A = 1/2 bh, Solve for h First we multiply both sides by 2 to get rid of the fraction. 2A = bh... now divide both sides by "b" to get "h" by itself. h = 2A / b S = P + Prt, solve for t Get the term with "t" in it all by itself on the right side. So, we need to subtract P from both sides. S - P = Prt... now we can divide both sides by "Pr" to get the "t" on the right (S-P) / Pr = t... that is the answer! If the equation contains the variable in MORE THAN ONE location then all terms containing the variable must be placed on the same side of the equation. Factor out the chosen variable, then divide by the remaining parts. : xy + 5 = x + 7, solve for x Move all terms with "x" to the left side. Move all other terms to the right side. xy - x = 7-5 xy - x = 2... now factor out the "x" on the left (this is the tricky part) x(y-1) = 2... finally, we divide both sides by (y-1) to get the solution for "x" x = 2 / (y-1) Now lets talk about Decimals and Percents. You must know that all percents can be written as decimals. For example, 10% is the same as 0.10 and 360% is the same as 3.6. To go from a percent to a decimal, move the decimal point for the percent number TWO places to the LEFT. For example, 2.15% is the same as 0.0215 Now, to convert a decimal to a percent, move the decimal point TWO places to the RIGHT. For example, 0.16 is the same as 16.0% or 16%. Now lets do some word problems. Use the formula A=PB, which is A is P percent of B. : 32% of what number is 51.2 So we are solving for B, where A=51.2 and P=32%=0.32 51.2 = 0.32*B.. divide both sides by 32 51.2 / 0.32 = B, then B = 160

MML Module #3 Section 2.5 An Introduction to Problem Solving This section focuses on solving word problems, which can be tough for many people. Recall from Section 1.1 that phrases can be written as algebra expressions. In each problem we will need to convert the written words into equations that can be solved. Lets try some examples... : The sum of 8 and 2 times a number is 10... 8 + 2x = 10 : A field is twice as long as it is wide. The perimeter is 90. What are the dimensions. HINT: The words say the field is longer than it is wide, so be sure your equation makes sense. Equation 1: 2W = L (notice that L is larger than W here) Equation 2: 2W + 2L = 90 (perimeter is all around the outside) Use substitution to solve... L + 2L = 100, 3L = 90, L = 30 If L=30, then 2W = 30 and W = 15. : Consecutive ODD integers whose sum is 28. X + (X+2) = 28 2X + 2 = 28 2X = 26 X = 13 and X+2 = 15 Hint: Notice that consecutive ODD or consecutive EVEN integers are always X and X+2 since the numbers are separated by 2. Section 2.6 Solving Linear Inequalities In most textbooks you will find a special notation for inequalities called Set-Builder Notation. For example: This means, the set of numbers X, such that X is less than 4. Note: Some books will show this as:, so don t let that confuse you. This means all of the numbers from NEGATIVE-Infinity to 4, which is the same as LESS THAN 4. Solving for inequalities is similar to Section 2.3, but there are a few subtle tricks. Try to get into a habit of leaving the variable on the left side and read things left to right. If the variable is on the right side, we must read the inequality right to left (sounds dyslexic). : 2 < x is the same as x > 2 If at any time you must divide by a NEGATIVE number then you must FLIP the inequality sign. This is a very important part of solving inequality equations. -7x > 21 divide both sides by -7 (requires you to FLIP the SIGN) X < -3 which is also

4(2y-1) > 12 8y 4 > 12 8y > 12+4 8y > 16 divide both sides by 8 to get: y > 2 or Chapter 4 Linear Equations and Inequalities in Two Variables MML Module #4 Section 4.1 Graphing Linear Equations in Two Variables The Cartesian Coordinate system is the X-Y plane where points (ordered pairs) can be plotted using the X and Y coordinates. The X-axis runs left-to-right and the Y-axis runs up-and-down. We plot an ordered pair (X,Y) but finding the X and Y coordinate points on the graph. So lets look at the following graph and discuss the ordered pairs: Ordered Pairs or Points on the Graph: A = (5,2), B = (-2,4), C = (-6,5), D = (-4,-1), E = (-2,-3), F = (0,-1), G = (5,-3), H = (2.5, -0.5) Quadrants: The coordinate system is broken into 4 quadrants. They are labeled 1,2,3 and 4. They start in the upper-right section (positive X and Y) for quadrant 1 and then rotate counter-clockwise to the left for quadrants 2 to 4. For the example above Quadrant 1: A, Quadrant 2: B, C, Quadrant 3: D, E, Quadrant 4 : H, G Notice that point F is not in any quadrant, since it falls on the axis. A Linear Equation has a single power in X and Y. The solution is the set of points (X,Y) that fit into the equation to form a line on the Cartesian Coordinate plane. Y = 2 X 3 To graph this line set up an X and Y table and choose at least 3 points (it only takes two to make a line but this makes sure there is no error in your calculations). Use the chosen X points and solve for the Y point that goes with each one. Graph the points and connect the line. HINT: It does not matter what you choose for X, there are an infinite number of points on a LINE, so pick some reasonable values for X to give you points with a reasonable value of Y that fit on your graph.

Graphing in My-Math-Lab (MML) can be tricky. Please download my example notes for "Drawing a Line in MML using Points". This will guide you through the process for MML graphing. Section 4.2 Graphing Linear Equations using Intercepts The intercepts are points where the graph crosses the X or Y axis. These points can be used to graph a line since it takes only 2 points to make a line. Here is how to solve for the intercepts... X-Intercept: Let Y=0 and solve for X Y-Intercept: Let X=0 and solve for Y 6X 2Y = 12 X-Intercept: Let y=0 then 6x=12 and x=2 Y-Intercept: Let x=0 then 2y=12 and y= -6 A horizontal or vertical line has a special equation: Horizontal: Y=constant, such as Y=3 Vertical: X=constant, such as X = -5 Section 4.3 Slope The slope measures the steepness of a line as it moves from left to right. Positive slopes go UP to the right and negative slopes go DOWN to the right. This is an important concept, so know how to find the slope of any equation, the slope of a graph, or the slope of a line given 2 points. Slope = m = where two points are (x 1,y 1 ) and (x 2,y 2 ) (6,-4) and (4,-2) Slope= Horizontal lines will have a Change of Y equal to zero and a Slope=0. Vertical lines will have a Change of X equal to zero which makes the slope equation undefined. Therefore, the slope of a vertical line is UNDEFINED. Parallel lines have the same slope. That is m 1 =m 2.

Word Problems... when finding slope for word problems, recall that slope is the "rise" or vertical change divided by the "run" or horizontal change. Slope of a Line on a Graph... when finding the slope of a line on a graph, just select any 2 points on the graph (x,y) and then calculate the slope using the equation above. Any 2 points will work, just be sure to get the values correct. MML Module #5 Section 4.4 The Slope-Intercept Form of the Equation of a Line There are 3 basic forms of line equations: Point-Slope, Slope-Intercept, and Standard. Standard Form: Ax + By = C This form is often used in math books. The constants A, B, C may be fractions or decimals. Given an equation in Standard Form you may solve for Y to get slope-intercept form for each graphing. This section deals with Slope-Intercept Form: Y = m X + B Slope (m) is calculated the same as above, but the B term is now the Y-Intercept. You can solve any linear equation for Y to get an equation with this form. Find the slope and intercept Y = -3/4 X + 6 Slope = -3/4, and Y-Intercept = 6 Find the slope and intercept 3Y = -9X Solve for Y: Y = -3X + 0 Slope = -3, Y-Intercept = 0 This line goes through the ORIGIN (0,0) and has a slope of -3. Section 4.5 The Point-Slope Form of the Equation of a Line Remember that Y and X are variables in the point-slope equation. We need to replace the coordinate point (x 1, y 1 ) in the equation. Slope = -11, Point (0,-3) Y Y1 = M (X X1) Y (-3) = -11 (X 0) Y + 3 = -11X Y = -11X 3 The next example is VERY important. You should know how to find the equation of a line given any 2 points. Points (3,6) and (-5,-2) Find Slope = 2 6 5 3 = 8 8 =1 Now, pick one of the points to use with the slope of m=1 lets use (3,6) Y 6 = 1 (X-3) Y 6 = X - 3 Y = X - 3 + 6 Y = X + 3

Find the equation of the line through (2, -4) and parallel to the line through the points: (6, 2) and (-2, 6). Section 4.6 Linear Inequalities in Two Variables There are two things you need to understand for this section. First is how to test a point in a given inequality equation. The second is how to graph an inequality. Determine whether the ordered pairs (0,5) and (1,8) are solutions to Y > -4x + 8 Replace each point in the equation... (0,5) 5 > -4(0) + 8 5 > 0 + 8 5 > 8... this is FALSE Replace (1,8) in the equation 8 > -4(1) + 8 8 > -4 + 8 8 > 4... this is TRUE Graph: 12 >= 2X - 3Y First solve the equation for Y so that we can use the Slope-Intercept form to determine the Y-intercept and then get the slope. 12-2X >= -3Y... divide both sides by -3, which means we must FLIP the SIGN 12/-3-2X/(-3) <= Y... flip over to other side Y >= (2/3)X - 4... slope = 2/3, y-intercept is (0,-4) HINT: When graphing in MML, use the LINE TOOL and the REGION SHADING TOOL for graphing the line and then shading one region of the graph for the inequality.

Chapter 5 Systems of Linear Equations and Inequalities MML Module #6 Section 5.1 Solving Systems of Linear Equations by Graphing The solution to a single equation, such as: Y = X + 2 is all of the X and Y points (X,Y) that fit into the equation. When we plot those on a graph, we get a straight line. The solution to a system (one or more) of linear equations (lines) is the set of points that satisfies ALL of the equations. So for example, lets take 2 lines. If these lines intersect in one point, that will be a single solution. If these lines are parallel and never intersect, then there is no solution. If these lines are the same and are on top of each other, then the solution is infinite, since there are an infinite number of points that will work. Linear equations are used often in basic mathematics and are also used to represent data within many science and business fields. It is essential that students know how to handle these equations. Any linear equation is basically a line that can be graphed on a 2-dimensional X-Y graph. So here are some examples. Given the equation: y = 2x 6 Determine if the following data points are solutions: (3,0), (2,-2), (1,2) In order to do this, we must take each data point (x,y) and plug it into the equation. If it checks out mathematically, then it is part of that line and known as a solution. Test (3,0) Test (2,-2) Test (1,2) 0 = 2(3) 6 = 6-6 = 0 YES -2 = 2(2) 6 = 4-6 = -2 YES 2 = 2(1) 6 = 2-6 = -4 since 2 does not equal -4, then NO We can solve a set of linear equations using graphing. Here are the steps: Step 1 Find 3 data points for each line. Just take X=1,2, and 3 and solve for each Y. This will make 3 data points (x,y) on the line Step 2 Plot both lines on the same graph. Step 3 The point of intersection (x,y) is the solution of this set of equations. Given: x + y = 4 and x + y = 2, find the solution Find at least 3 data points for each line. I think it is best to make a table for each one and then graph the data points when you are done. Notice the intersection point is (1,3). This is the solution for these equations.

There are 3 possible types of solutions. One solution, where the 2 lines intersect No solution, where the lines are parallel and never intersect Infinite solutions, where the lines are identically the same In the case of the infinite solution, you may have a case like this: Equation 1: x + y = 2 Equation 2: 2x + 2y = 4 Notice how the second equation is just 2-times the first equation. If you take the second equation and divide every term by 2, you get the first equation. So, if we graph these 2 equations, they will be the very same. There are an infinite number of points or solutions that match these two lines. Section 5.2 Solving Systems of Linear Equations by Substitution Method Section 4.1 presented the graphing method for solving a set of equations. It is also possible to solve them using algebra. Here is a method called The Substitution Method Step 1 Choose one equation and solve it for a single variable (X or Y) Step 2 Substitute the value of the single variable into the other equation. This will leave you one equation with only ONE variable. Step 3 Solve that equation for the ONE variable. This is done by moving all of the variables onto one side and the numbers on the other side. Step 4 Substitute that solution back into the original first equation and solve for the value of the other variable. You will be left with a single (x,y) solution. : x + 4y = 7 and 2x + 3y = 5 Step 1 First, choose the one equation that is easiest to solve for a single variable. It looks like the first equation can be solved for X, pretty easily. It is: x = -4y + 7 Step 2 Substitute x = -4y + 7 into the x of the second equation. 2 (-4y + 7) + 3y = 5. Now expand and group similar terms -8y + 14 + 3y = 5

-5y + 14 = 5-5y = -9, or y = 9/5 Step 3 Substitute y = 9/5 back into one of the original equations. Step 4 x + 4(9/5) = 7.. solve for x x + 36/5 = 7.. then x = -36/5 + 7 = -36/5 + 35/5 = -1/5. So the total solution (x,y) = ( - 1/5, 9/5 ). You can check your solution is correct by substituting this answer into the other equation. Try it out. : Y = 3(X-4) Use the substitution method to solve 3X Y = 12 Using the steps for Substitution, the first equation is already solved for one variable, Y. Plug that into the second equation and solve. The first equation is expanded to be: Y = 3X - 12 3X (3X-12) = 12. 3X 3X + 12 = 12. 12 = 12, or 0 = 0. This solution is INFINITE, or each other.. If you graph the two lines, they are the same and will lie on top of IMPORTANT... if all of the variables cancel out and you are left with 2=2 or 0=0, etc. then the solution is INFINITE. In the other case, if you are left with 0=2 or 1=4 where the numbers do not match, then we have NO SOLUTION. An example of NO SOLUTION is in the next section. 5.3 Solving Systems of Linear Equations by Addition Method Another method to solve a set of linear equations is to use the Addition Method. This method combines steps 1, 2 and 3 of the Substitution Method to solve for one variable. We then follow steps 4 and 5 to get the final solution and check the result. By the way, if given a choice, you are free to use any method you desire to solve these equations. The key is that you get the correct solution!!! The process of the Addition Method is best explained by completing a few examples. Here are the general steps: Addition Method 1. Write both equations in the same form. For example: x+y=5, etc. In some cases, we may want to eliminate fractions by multiplying by the Least Common Denominator (LCD) or eliminate decimals by multiplying by a large number. 2. Multiply (one or both) equations by numbers (if necessary) to get a positive and negative value for one variable so that it can be eliminated when we add the equations. This part can be tricky, so look at the example below. 3. Add the equations (vertically) and solve for the one variable. 4. Substitute that value back into one of the original equations to solve for the other variable. Make sure you use the original equations, since you may have made an intermediate error. 5. Check your solutions. : x + y = 6 and -2x + y = -3 First, we must place the equations on top of each other and then see if we can add them together and get a variable to cancel out. In our case, we could add the x and -2x column or the y and y column, but neither of these removes a variable. So lets multiply one the equations by a number to make this work. So we choose to multiply the top equation by -1 so the y terms will cancel. -x - y = -6-2x + y = -3 now add the 2 columns

=========== -3x = -9 notice how the y-terms cancel out Solve -3x = -9, then x=3. Place this into equation 1 to get: 3 + y = 6, or y=3. So the overall solution is: (3,3). Check this in the second equation to be sure!!! : 3x - 4y = 11-3x + 2y = -7 --------------------- We want to add these two equations in the vertical direction in each column to eliminate ONE variable. Notice that this equation is already in perfect form to eliminate the X variable. When we add 3x to 3x we will get ZERO in the first column. Here is the result when we add: 3x - 4y = 11-3x + 2y = -7 -------------------- 0-2y = 4 Solve this to get Y = -2 Now substitute back into one of the original equations. 3x 4(-2) = 11, then 3x + 8 = 11, 3x = 3, X=1 The solution is X=1 and Y= -2, or (1,-2). How do we check this solution??? This is an important step that you should do EVERY time. I like to use the equation that was NOT used in the last substitution step. So take our solution and test it into equation #2. -3 (1) + 2 (-2) = -7??? -3 4 = -7, the 7 = -7 OK, it all checks out!!! : 3x + 4y = -5 Solve by the addition method 5x + 6y = -7 In order to solve the addition method, we need to eliminate one variable. In the current form, this is not possible, since when we add vertically nothing is eliminated. We will need to multiply each one or both equations by numbers to change their format and allow us to use the addition method. If I multiply equation #1 by 5, we will get 15x for the first term. If I multiply equation #2 by +3, we will get 15x for the first term. When I add them vertically, it will eliminate the x-term, which is our goal. Note: We could choose to eliminate the y-terms by choosing multipliers of 6 and 4 in the same way. In any case, when we use the multipliers, be sure to multiply EVERY term by that number to change the equation. The equation is still the same, but just in a different form. 3x + 4y = -5 Multiply by 5 to get: -15x - 20y = 25 5x + 6y = -7 Multiply by +3 to get: 15x + 18y = -21 ============== Now add the two equations vertically 0-2y = 4 Solve to get: Y = -2 Substitute into the first equation to get: 3X + 4(-2) = -5 3X 8 = -5. 3X = 3, X=1 Solution is: (1,-2) : Solve using the addition method

First, I like to eliminate fractions by multiplying each equation by its LCD. This gets rid of all fractions and makes it easier at least I think so. Equation 1 multiply by LCD=3 to get: 3y = -2x 9 Equation 2 multiply by LCD=2 to get: 2x = -3y + 18 Now arrange to get in a common order Ax + By = C 2x + 3y = -9 2x + 3y = 18 To eliminate the x-variable from the equations, we will need to multiply the second equation by 1 so that we have a 2x and a -2x to add and eliminate. 2x + 3y = -9-2x - 3y = -18 Multiplied each term by -1 ============= 0 + 0 = -27 Add equations vertically 0 = -27 Notice that all variables are removed and the numbers are NOT the same. This is NO SOLUTION. The book will use the empty set or NULL SET to represent this case. 5.4 Problem Solving using Systems of Equations Word problems with multiple sets of equations can be difficult. Carefully use the words to determine the equations and then use the skills from Sections 5.1 to 5.3 to solve the equations. These just take PRACTICE. : You are choosing between two plans. Plan A offers an annual membership of $130 and you pay 40% of the list price. Plan B offers an annual membership of $40 and you pay 90% of the list price. How much would you have to purchase in a year to pay the same amount under both plans? What is the cost for each plan? TOTAL COST = Annual Cost + Rate * (List Price) Plan-A Cost = 130 + 0.40 * X... where X = List Price Plan-B Cost = 40 + 0.90 * X 130 + 0.40(X) = 40 + 0.90(X) 130-40 = 0.90(X) - 0.40(X) 90 = 0.50(X)... divide both sides by 0.50 90 / 0.5 = X X = 180... This is the total amount of all merchandise at the List Price Plan Fee = 130 + 0.40(180) = 130 + 72 = 202... this is the Cost of each Plan

Chapter 6 Exponents and Polynomials MML Module #7 Section 6.1 Adding and Subtracting Polynomials Know the pieces of a polynomial by placing highest powers first (on the left) and the lower successive powers to the right. Order or Power = 3 (highest power term) Leading Coefficient = 2 (constant in front of the highest power term) Constant Term = -7 (last term with no variable, including the sign) When adding or subtracting polynomials we can only add terms that are similar (same powers). (y-2) (7y-9) = y-2-7y+9 = -6y+7 Section 6.2 Multiplying Polynomials Exponents are often used for repeated multiplication : When multiplying two exponents with the SAME base, we ADD exponents... notice it is NOT Product to Powers Rule: An exponential POWER outside of a set of parenthesis is applied to EVERY term inside the parenthesis. If powers already exist inside the parenthesis, we can multiply those powers. Multiplying Monomials: Multiply the coefficients (numbers) and then multiply the similar variables together. Remember if the bases are the same for the variable exponents, then we ADD the exponents.

Multiplying Binomials: Binomials are polynomials with TWO terms. When multiplying we can use the famous FOIL Method to multiply the first terms (F), the outer terms (O), the inner terms (I), and the last terms (L). Then we combine similar terms to simplify. In most cases, the O and I terms will combine. Note: This is the same thing as taking each term on the left say the X and 4 and multiplying them into the term on the right (x-6). You get 4 pieces in the second step. Binomials and Trinomials: These more complicated multiplications need to be done in pieces. Take each term on the left and distribute it to the term on the right. Go one step at a time. See if you can follow this one I started with the y term multiplied by every term in the right parenthesis Section 6.4 Polynomials in Several Variables Adding, subtracting and multiplying polynomials with several variables can be confusing. Be very careful to only combine terms with the same exact type of variables. Also be careful of powers. There are no other great tips to add. The best approach is to follow some examples like below. (5x 2 y 3xy) + (2x 2 y xy) 5x 2 y + 2x 2 y 3xy xy 7x 2 y 4 xy (10 x 2 y)(5xy) 10*5*x 2 * x * y * y 50x 3 y 2 Notice how similar terms (variables) were combined Multiply similar terms and combine

Section 6.7 Negative Exponents and Scientific Notation In Section 1.8 we discussed exponents. : Negative exponents should be converted to positive exponents by flipping over the term in fractional form and changing to a positive exponent. : : When multiplying two exponents with the SAME base, we ADD exponents (Product Rule). Any exponential term with a ZERO exponent always equals ONE (1): When dealing with fractions there are several tricks to solving and reducing. Moving terms from top to bottom (numerator to denominator) or from bottom to top (denominator to numerator) will change the sign of the exponent. This is helpful to put exponents on the same level for combining by the power rule. When powers are combined in fractions we call this the Quotient Rule. Finally, NEVER leave negative exponents in the final answer. Always eliminate them by moving their position in the fraction. Here are some additional examples: First simplify the top and bottom separately. Remember to add exponents when the bases are the same in this case we add the exponents of the two X terms on the top. Next, move any negative powers to convert them to positive powers. For instance, a negative power on the top of a fraction is the same as a positive power on the bottom. First used product rule on each part of the fraction, second moved the y-terms to the same level in the fraction (top) to combine using the power rule, and last moved the exponent number terms to make positive.

Changed negative exponents to positive by flipping over Now lets talk about Scientific Notation. We can basically take any decimal or whole number and put it into scientific notation. The form is the following... Number = A x 10 n, where A is between 1 and 10, and "n" is an integer. To convert a number into Scientific Notation we first move the decimal to a point where we have a number between 1 and 10 and then we count the number of place we moved the decimal. If we moved the decimal to the right, then n<0. If we moved the decimal to the left, then n>0. Here are some examples. Convert 32400 into Scientific Notation Start with 32400 (decimal is after the last ZERO). We would move the decimal 4 places to the left to get 3.24 (number between 1 and 10). Solution: 3.24 x 10 4 Notice that 10 4 is the same as 10*10*10*10 = 10,000 If we do the math it works... 3.24 * 10,000 = 32400. Convert 2.15 x 10-2 into a decimal We can do this two ways... first, we know that n<0 so to get back to a decimal, we need to take 2.15 and move the decimal place 2 places to the left. Since 10-2 is smaller than 10, this makes sense since our answer is a smaller number than 2.15. So, we get 0.0215. Another approach is to think that 10-2 is the same as 1 / 102, or 1/100. So we get... 2.15 x (1/100) = 0.0215 MML Module #8 Chapter 5 Section 5.5 Systems of Linear Equations and Inequalities Systems of Linear Inequalities Solving systems of linear inequalities is nearly the same as solving systems of regular equations like we did in the other sections of Chapter 5. The only difference is that we will not end up with a single point solution, but rather a region of points. The key is to graph each line separately and then mark the shading based on the inequality. The common region where the two shaded sections cross, is our solution. Here is an example from MML. Graph the solution of these linear inequalities... X + Y > 1 and X + Y < 5. X + Y > 1... solve for Y to get... Y > -X + 1 X + Y < 5... solve for Y to get... Y < -X + 5 To graph these lines in MML, you need to use several tools. Select the LINE tool, the SHADING tool and then the LINE TYPE tool. If you have < or >, we will use a DASHED line, which means those points are

NOT part of the solution. If you have <= or >=, we will use a SOLID line, which means those points are part of the solution. That tool is on the far right of the MML graph tool box. Chapter 10 Quadratic Equations and Introduction to Functions Section 10.6 Introduction to Functions Functions are relations between a set of data points (ordered pairs). The first part of the data point is the DOMAIN and the second part of the data point is the RANGE. Typically, the x-coordinates are the DOMAIN and the y-coordinates are the RANGE. To be a FUNCTION, each part of the DOMAIN must have only ONE corresponding part in the RANGE. So, if the points are (1,2) (2,3) and (3,3) that is a FUNCTION since each DOMAIN... which is 1,2,3 has only one RANGE value, even though 3 is repeated twice in the range. Now, here is another example. (1,2) (2,3) (1,3). This is NOT a FUNCTION. Since the domain value x=1 has a range value of both 2 and 3, it is NOT a function. Another way to test for a FUNCTION is the "Vertical Line Test". If you can draw a vertical line through a graph and it touched the curve MORE than one time, then it is NOT a function. Here are some examples:

#20... this graph is a FUNCTION #22.. this graph is NOT a function since a vertical line crosses the curve two times. Evaluating a function is anther important skill. First we need to understand function notation. f (x) = x 2 + 3x 4 This notation means the "f" is a function of the variable "x" according to the given equation. If we want to find f(3), or the function evaluated at x=3, then we substitute x=3 into every x-variable in the equation. f (3) = 3 3 + 3(3) 4 = 9 + 9 4 =18 4 =14 Finally, if we need to identify the DOMAIN and RANGE from a graph, we need to find the total limit of the x-coordinates for the DOMAIN and total limit of the y-coordinates for the RANGE. Lets look at this example: See graph on the right... The x-values (DOMAIN) start at x=-1 and it goes to the right. the x-values go out to infinity, so we say the DOMAIN is x >= -1 The y-values (RANGE) start at y=0 and go up to infinity as x gets larger. The RANGE is then y >= 0.