3.1 Solving Systems Using Tables and Graphs

Transcription

1 Algebra 2 Chapter 3: Systems of Equations Mrs. Leahy 3.1 Solving Systems Using Tables and Graphs A solution to a system of linear equations is an that makes all of the equations. To solve a system of equations by graphing: 1. each line very carefully. 2. for where the lines intersect. 3. the solution in both equations. Example: Solve each system by graphing 1. y 2x1 1 y x y x y x 1 4 1

2 Algebra 2 Chapter 3: Systems of Equations Mrs. Leahy 3. 8x 4y 12 y 2x 3 2x y 3 y 2x x y 6 y 7x 6 7x y 6 y 7x 6 So what words do we use to describe these systems? Consistent: has at least solution Inconsistent: has solution Independent: has solution Dependent: has solutions. 2

3 3.2 Solving Systems Algebraically Sometimes, it is difficult to graph equations accurately by hand. Sometimes, the points of intersection are not integer values and are difficult to see. Sometime, we need a different way to find the solution to a system of equations. A: Solving Systems by Substitution Example: Solve by substitution x 2y 5 1. x 2 3y Make a Plan: This method works well when One equation is already solved for one variable in terms of another like y= or x = One equation has a term with a coefficient of 1 or -1. Make it Happen: If necessary, solve one equation for one of the variables. Get x = or y = 2. y2x6 7x 6y 1 Substitute the expression for the variable in the other equation. Solve the equation. You will almost always have to distribute first. Substitute your solution into either original equation. Solve for the remaining variable. Write your answer as an ordered pair. CHECK YOUR ANSWER IN BOTH EQUATIONS! 3

4 (Solve by Substitution, Continued) 3. 4x3y 10 x 2y x y 12 3x7 y 1 5. y 2x1 1 y x x2y14 2x2y 6 4

5 B: Solving by Elimination 3x2y x2y 6 Make a Plan: This method works well when No variable is already solved for No coefficient is 1 or -1 Make it Happen: GOAL to create additive opposites Ex. 4x, -4x or -7y, 7y If necessary, multiply one or both equations by a constant to create additive opposites. Add the two equations. This should eliminate one of the variables. 8. 4x3y24 4x 7y 36 Solve for the variable. Substitute into either original equations and solve for the remaining variable. Write your answer as an ordered pair. CHECK YOUR ANSWER IN BOTH EQUATIONS! 9. x2y 10 3x y 9 5

6 10. 2x7y4 3x 5y x3y18 7x2y x y 2 2x2y x y 6 12 x3y18 6

7 3.5 Systems with Three Variables In lesson 3.1 & 3.2, we learned about systems of 2 equations with 2 variables, that may have a solution (x, y) that makes both equations true. In this lesson, we will learn about systems of 3 equations and 3 variables that may have a solution (x, y, z) that makes all three equations true. Example: Solve each system of equations. Check your answers. Make a Plan GOAL: Use elimination (lesson 3.2) to change your 3 x 3 system to a 2 x 2 system. 1. x y z 1 x y 3z 3 2x y 2z 0 Look for a variable that can easily be eliminated from all three equations. Make it Happen Choose a variable to eliminate. Choose 2 equations & eliminate the variable. Choose a different pair of equations & eliminate the SAME VARIABLE again. You now have TWO new equations each with TWO variables. Use substitution or elimination to solve (Like 3.2) Substitute your solutions into one of the ORIGINAL 3 variable equations and solve for the missing term. Write your answer as an alphabetical ordered triplet. (x, y, z) 7

8 2. x y 2z 7 3x y 2z 7 x 3y z 9 3. x 2y 3z 12 2x y 2z 5 2x 2y z 4 8

9 3.6 Using Matrices to Solve Systems of Equations We will be using a graphing calculator to do the major computation in this section. On the test, you will be require to solve problems with the graphing calculator on your own, so be sure to pay close attention today! About Matrices A matrix is a rectangular array of numbers, symbols, or expressions. The individual items in the matrix are called elements or entries. The matrix at the right has elements The size of a matrix is given by the number of rows 1 st and then the number of columns second. The matrix at the right has rows and columns. We say the size of this matrix is:. The elements in a matrix are named a i, j where i = row position and j = column position. Name the elements: a 1,3 = a 2,1 = Augmented Matrices An augmented matrix can be used to represent a system of equations. It is important that each equation in the system be in standard form ( Ax By... D ), that is, all the variable terms are on the left side of the equals sign in alphabetic order and the constant term isolated on the right side of the equals sign. System of Equations in Standard Form 5x 2y z 5 3x 3y 3z 9 x 2y 4z 6 Augmented matrix Write the following system of equations as augmented matrices: 1. x y z 6 2x 3y 2z 2 3x 5y 4z 4 2. x 2y 10 2x 3y 1 9

10 Reduced Row Echelon Form We are going to use a graphing calculator to change the augmented matrix for a system of equations into the reduced row echelon form (RREF) of the matrix. From RREF, we will be able to read the solutions of the system of equations that it represents. Example of RREF: Solutions: Read the solutions off the following RREF matrices: TI-83 Matrix Menu MATRX 2 nd x -1 About the Matrix (MATRX) menu: NAMES used to paste the name of a matrix into the home screen or into a program. MATH contains all the operations that can be done with a matrix. EDIT where you set the size of the matrix and enter the elements. *NOTE the TI 83 may have a Matrix button instead of a Matrix you have to shift to use. This is actually easier, because it is one less step to make. 10

11 Example: SOLVE the following system 5x 2y z 5 3x 3y 3z 9 x 2y 4z 6 INSTRUCTIONS 1. Write your system as an augmented matrix and determine the size. 2. Go to MATRX and EDIT. ENTER to select [A] 3. Define [A] to be a 3 x 4 matrix Use the cursor and ENTER to move through the spaces. 4. Enter the coefficients and constants from the system (your augmented matrix) by entering the value and pressing ENTER. 5. Return to the home screen 2 nd MODE to QUIT 6. Go to MATRX and MATH. Scroll down and hit ENTER to select B: rref( 7. Go to MATRX and NAMES. Hit ENTER to paste [A] to your home screen. Close the parenthesis. 8. Press ENTER and read the solution from the matrix. SOLUTION: x = y = z = 11

12 Use a graphing calculator to solve the following systems of equations x3.25y7 3x7.5y0 2. 2x 5y 11 x y x y 3z 15 3x 4y 2z 7 2x 2y 5z x 4y 3z 3 x 2y 7z 5 3x 5y 3 Day 2: Set up a system of equations and solve using a graphing calculator. Does your answer make sense? 1. The tickets for the school play are $7.00 for adults and $5.00 for students. There were 100 tickets sold. If the total profit was $550.00, how many of each were sold? 12

13 2. You work at a fruit stand that sells apples for $2 per pound, oranges for $5 per pound, and bananas for $3 per pound. Yesterday you sold 60 pounds of fruit and made $180. You sold 10 more pounds of apples than bananas. How many pounds of each kind of fruit did you sell? 3. Suppose you want to fill nine 1-1b tins with a snack mix. You have $15 and plan to buy almonds for $2.45 per pound, hazelnuts for $1.85 per pound, and raisins for $0.80 per pound. You want the mix to contain an equal amount of almonds and hazelnuts and twice as much of the nuts as raisin by weight. How many of each ingredient should you buy? 4. A hardware store mixes paints in a ratio of two quarts red to six quarts yellow to make two gallons of pumpkin orange. A ratio of five quarts red to three quarts yellow makes two gallons of pepper red. A gallon of pumpkin orange sells for $25 and a gallon of pepper red sells for $28. Find the cost of a quart of red and a quart of yellow. 13