Algebra (Linear Expressions & Equations)

Transcription

1 ACT Mathematics Fundamentals 1 with facts, examples, problems, and solutions Algebra (Linear Expressions & Equations) One might say that the two main goals of algebra are to 1) model real world situations and also to 2) solve these equations for unknown values. We are going to show you how to do both. First, though, it s important to know one of the most standard equations: y = mx + b. This, you might remember (you use it every day, right?), is called slope-intercept form. Outside of the context of a graph, though, this models a very common progression in life: b - starting amount x - independent variable, like "months" m - gain or loss per x ("per month") y - dependent variable, ("total after x months") For example, nearly every person who owns a cell phone has encountered this at some point. Mrs. Washington, you will owe us $100 today for the phone, and your bill will be $8 per month. This translates into the equation y = 8x Mrs. Washington pays $100 up front, and $8 per month, where x is the number of months. Well, my son s been good this year, but how much will this phone cost me this year? Mrs. Washington could just plug in 12 (12 months = 1 year), and get , which would equal $1,120 (you might not be getting that phone). This seems simple, right? That s because it is. What the ACT does, and will do, though, is mire this in words like a truck in mud. This is what a question about Mrs. Washington might look like: Mrs. Washington is looking at buying a new phone from Phone World for her son for Christmas. Phone World charges a $100 fee on the day of buying the phone, and $8 per month for their standard talk and data plan. If m is the number of months after buying the phone, and p is the price after m months, which of the following equations could Mrs. Washington plug in 12 to calculate the cost of the phone and the standard talk and data plan after a year? F. p = 100m + 8 G. p = 8m H. 100p = 8m J. p = 8m K. p = 12m That s ridiculous, right! Tough word problem! y = mx + b

2 NO. It s a simple situation, one that you have probably dealt with yourself and could do with a calculator and seconds of time. The trick is to wade through to see what the problem really is: Mrs. Washington is looking at buying a new phone from Phone World for her son for Christmas. Phone World charges a $100 fee on the day of buying the phone, and $8 per month for their standard talk and data plan. If m is the number of months after buying the phone, and p is the price after m months, which of the following equations could Mrs. Washington plug in 12 to calculate the cost of the phone and the standard talk and data plan after a year? That looks much nicer, doesn t it? $100 up front, $8 per month J. p = 8m Done. What made that much easier was knowing what slope-intercept really means and UNDERLINING. Now you try that same underlining strategy and your knowledge of slopeintercept on a couple ACT questions. (Answers provided on final page.) 1. The employees of two factories, X and Y, are comparing their respective production records. Factory X has already produced 18,000 units and can produce 120 units per day. Factory Y has produced only 14,00 units but can produce 1 units per day. If d represents the number of days (that is, days during which each factory is producing its maximum number of units), which of the following equations could be solved to determine the number of days until X[ s total production equals Y s total production? A. 18, d = 14,00 + 1d B. 18, d = 14, d C. 18, d = 14, d D d = 18,000 14,00 E d = 18, ,00 2. Joe rents a car to drive across the state to visit his family for Thanksgiving. The car rental company charges Joe $112 for the weekend rental, plus $0.99 for each mile he drives. If Joe drives the rental car m miles, then which of the following expressions gives Joe s total cost, in dollars, for renting the car? F. 0.99m 112 G. 0.99m H. 49.9m J. 112m K m Now try a slightly more difficult question. 3. Lucy is studying her ant farm. She needs to approximate the number of ants in the population, and she realizes that the number of ants, N, is close to 0 more than double the volume of the ant farm, V. Which of the formulas below expresses that approximation? A. N 6 V B. N 6 V C. N 2V + 0 D. N 2 V + 0 E. N V 7 + 0

3 Here, the setup was a bit different. The slope was not really a slope at all. Nonetheless, the approach is the same. 0 more than implies something plus 0. Double the volume would be 2 times the volume, or 2V. Therefore, your answer is C. A final example: 4. An integer, x, is subtracted from 6. That difference is then multiplied by 3. This product is 1 more than half the original integer. Which of the following equations represents this relationship? A. 3 6 x = : B. 3 6 x + 1 = : 7 C. 3 6 x = 1 : 7 D. x 6 3 = 6= 7 E = : The big key here is knowing that is 1 more than half is + 1 = 1/ Now, let s talk about solving with algebra. First up, the distributive property. The distributive property allows you to distribute a multiplier across terms being added or subtracted within parentheses. That doesn t sound very nice, though, does it? Let s rephrase. Say you are buying fruit (or whatever). In each bag of fruit, there are apples and 6 oranges. That s 11 pieces of fruit. Let s say you buy 4 bags. How much fruit do you get? Let s start basic. 4 bags of (A + 6O) color coordinated, that's right 4*(A + 6O) 4 bags of means 4 times 4(A + 6O) Multiplication doesn't require a sign We all know you get 44 pieces of fruit. 20 apples, 24 bananas. Right? Let s delve deeper, though. There are two ways of looking at this. 4(11F) 44F F for fruit 44 pieces of fruit This seems elementary and easy, but pay attention to the fact that the two outcomes were the same. Adding within the parentheses and THEN multiplying gave the same value as multiplying by each of the values in the parentheses and then adding. Now for a reality check: The expression 9 y + 3 2(4y 4) is equivalent to: A. y 1 B. y + 1 C. y + 18 D. y + 23 E. y + 3 4*A + 4*6O 4 of each - the 4 distributes 20A + 24O 44 pieces of fruit

4 Maybe you crushed this in school. Maybe this is easy. For many, though, it s not. Here s the simple solution. 9 y y 4 = 9y y + 8 = y + 3 You now have 1 of 2 reactions. Either, Yeah, duh. Or, Why is it + 8 instead of 8? Here s why. 2(4y 4) says subtract 2 times (4y 4) So, subtract (8y 8) Subtracting 8y is well, 8y Subtracting 8, though is ADDING 8 In fewer words, the negative sign distributes, too. This is hugely important, and the ACT will test you on it. Here s a couple things that might help you. I'm not not mad basically means I'm mad. -8 might mean I owe 8 dollars. This is bad. -(-8) = 8 means You took away my debt. This is good. In short, subtracting a negative is a double negative, which is, of course, positive. Try these now.. For all real values of y, y =? F. 2y 9 H. 2y + 8 J. 2y 1 J. 2y K. 2y If 4 w 2 w = 46, then w =? A. 8 B. 10 C. 16 D. 18 E Which of the following is equivalent to the expression F. G = G. G 6I H. J G J. FJCF76 GCD6= K. 7CFE HCD6J 7 CDE FG =DH CDE?

5 8. If 4 x + 9x = 8, then x =? A. B. 8 C. 9 D. 10 E. 13 Now let s talk solving with algebra and order of operations. PEMDAS Please Excuse My Dear Aunt Sally Parentheses Exponents Multiply Divide Add Subtract In arithmetic (without variables) and algebra (with variables), this says that things inside parentheses happen first, then exponents, and so on. Here s an example of that come to life. 9 y 13 = 16 This is what is actually happening to poor old y: Subtract 13 y 13 Multiply by 9 9(y 13) Now, to solve this equation, simply reverse the steps with inverse operations. Divide both sides by -9 y - 13 = 16-9 Add 13 to both sides y = = Notice how the problem was solved from bottom to top. The LAST step was negated by the FIRST step of the solution. Let s look at another. What is the value of x when 4x + 7 = 6? Here, three things are happening to x. Multiply by 4 Divide by Add 7 Again, we just reverse the process. Go BACKWARDS, and do the OPPOSITE operation. Here s one for you to try (don t worry, there will be more practice later). 4x -9 4x 4x Subtract 7 from both sides = -1 Multiply by 4x = - 4x + 7 Divide by 4 x = - 4 9

6 9. What is the solution to the equation 9x 3x 1 = 3? F. 3 G. 7 E H. 6 E J. 7 E K. 3 Slope-intercept Form We re now going to use some of the algebra we just worked on to solve very common ACT questions regarding slope-intercept form, which we have worked with already. Now, however, we are going to specifically use m as slope and b as the y-intercept. Example: In the standard (x, y) coordinate plane, what is the slope of the line with equation 7y 3x = 21? Using our algebra skills, all we need to do is get this equation in the form y = mx + b. 7y 3x = > Add 3x to both sides 7y = 3x > Divide both sides by 7 The slope of the line is therefore 3. y = 3 x > m = 3 7 Try one yourself now: 10. What is the slope of the line represented by the equation 10y 16x = 13? A. 16 B. 6E 6I C. U = D. 10 E. 16 Slope is commonly calculated as rise over run, or the formula m = (V WFV X ). Here s an example. (: W F: X ) What is the slope of the line that goes through the points ( 2, 4) and 3,? Solution: = = 9 Try one yourself. 11. In the Cartesian plane, a line runs through points (1, ) and (, 10). Which of the following represents the slope of the line? F. H 6= G. H = H. 1 J. = H K. 6= H

7 Statistics & Probability The ACT also includes a decent amount of statistics questions, typically relating to one of three measures of a set of data: mean, median, and mode. Let s tackle all three. 1. Mean the average of a set of numbers; add the numbers up and divide by how many numbers there are in the set 2. Median the literal middle number; the number that splits the set into two equal halves 3. Mode the number that appears the most in the data set Suppose we have the following data set : A = {, 2, 9, 12, 4, 3, 0, } In general, it is always good to ORDER the set. If we reorder our set, we get: A = {0, 2, 3, 4,,, 9, 12} This representation helps us find the median and mode quickly and easily. The mode of A is clearly, since it shows up the most in A. To get the median, we need the number in the smack middle of the set. Since this set has 8 numbers, the middle number won t actually be in the set; it will be halfway between 0 and 3, so: Median of A: 4 + = 4. or Now let s get to the mean. This is by far the most important part of this section. To obtain the mean, we first need to determine the overall amount of the set: = 40 Keep this total in mind! To get the mean, we now just divide our total, 40, by : Mean of A: 40 = 8 Let s see a visual picture of this. Total: 40 Notice that a set of 8 with a mean (average) of says that the set must have a total of 40. Total: 40 Evenly distributed

8 Now let s look at an ACT-style question which requires this kind of understanding. 12. Kate and Shawne are doing a science project for Mr. Washington s science class. Their project goal is to show that the average growth of pea plants after a month is 6 inches given a certain kind of soil. Kate and Shawne have 9 plants. If the growths in inches after a month are, 2, 9, 12, 4, 3, 0, and for the first 8 plants, how much will the 9 th plant need to have grown to achieve the project goal? A. 6 B. 7 C. 9 D. 10 E. 14 Solution: Kate and Shawne need an average of 6 inches for 9 plants. We already figured out above that the average for the first 8 plants (the data set is the same) is only inches. So, clearly, the answer needs to be above. If you think about it, though, the 9 th plant will not only need to be at least 6 inches, it will have to compensate for the other plants. Let s go to a picture: Total: 40 +??? 12?? If the average needs to be 6 for all 9 plants, that means that the total needs to be 6 9 = 4. If the other 8 plants growths add up to 40 inches, then the missing length is 14 inches. Therefore, the answer is E. 6 Total: 4 Evenly distributed ? Plants #1-8 #9 The diagram on the right is what made this question relatively simple. The key on the ACT, though, is to understand the arithmetic carried out. You don t need to draw out a diagram to answer the question. Try the next two on your own.

9 13. Barry wants to bring his bowling average to a 10 with his last game. If his average in the first five games was 102, what must he score in his sixth and final game to meet his goal? F. 10 G. 110 H. 11 J. 120 K The set of four integers {m, m, n, p} has a mean of 0. Which of the following must be true? A. m = n B. n = p C. n + p = 2m D. n + p = 0 E. n + p = m

10 Answers to Practice Questions 1. A 2. G 3. C 4. B. J 6. D 7. K 8. E 9. H 10. C 11. K 12. E 13. J 14. C