THE SAT MATH TEST: THE HEART OF ALGEBRA
Skill 2: Working with Linear Equations
Lesson 5: Constructing and interpreting linear equations
The Horizon Resort charges $150 per night for a single room, and a one-time valet parking fee of $35. There is a 6.5% state tax on the room charges, but no tax on the valet parking fee. Which of the following equations represents the total charges in dollars, c, for a single room, valet parking, and taxes, for a stay of n nights at The Horizon Resort?
A) c = (150 + 0.065n) + 35
B) c = 1.065(150n) + 35
C) c = 1.065(150n + 35)
D) c = 1.065(150 + 35)n
(Medium) This question asks us explicitly to set up an equation to express a mathematical relationship in a word problem. Usually, this is just the first step in analyzing the situation more deeply, for instance, finding particular values of the variables that satisfy certain conditions, or interpreting the meanings of terms or coefficients in the equation, but this problem only asks us to set up the equation.
When translating verbal information into an equation, it’s helpful to take small steps. First, since the room charge is $150 per night, the charge for n nights is $150n. If a 6.5% tax is added to this, the room charge becomes 150n + 0.065(150n) = 1.065(150n). The $35 valet parking charge is added separately, and not taxed, so the total charges are 1.065(150n) + 35, and the correct answer is (B). Notice that this equation shows a linear relationship between c and n.
When setting up equations from word problems, try to classify the relationship (that is, linear, quadratic, exponential) between the variables, so that you can check that the equation is of the correct form. In this lesson, we will focus only on linear relationships, that is, relationships that can be expressed in the form y = mx + b.
Which of the following represents the equation of the line with an x-intercept of 6 that passes through the point (4, 4)?
B) y = 2x − 4
C) y = −2x + 12
D) y = −2x + 6
(Easy) This question asks you to construct the equation of a line given some facts about its graph. Start by drawing a graph (on the xy-plane) of the given information in the space next to the question. It also helps to know something about the different forms of linear equations and what they reveal about the graph of the line.
Graph of a line in the xy-plane
Forms of linear equations
Slope-intercept form: y = mx + b
Standard form: ax + by = c
Point-slope form: y − y1 = m(x − x1)
Features: slope = m, y-intercept = b
Features: slope = −a/b, y-intercept = c/b, x-intercept = c/a
Features: slope = m, point on line = (x1, y1)
In this problem, we are given two points on the line: (4, 4) and the x-intercept (6, 0). We can calculate the slope using the slope formula above: slope = (4 − 0)/(4 − 6) = (4)/(−2) = −2. If we use this slope and the point (6, 0), we can set up the equation in point-slope form:
Point-slope form of equation:
y − 0 = −2(x − 6)
Simplify and distribute:
y = −2x + 12
This is the equation in (C). Notice that this equation is in slope-intercept form, and reveals that this line also has a y-intercept of 12. Check this fact against your diagram, and also check that both given points, (4, 4) and (6, 0), satisfy this equation.
Lesson 6: Solving equations with the Laws of Equality
If , what is the value of x?
(Medium) At first glance, this doesn’t look like a linear equation. But one simple move reveals that it is:
Multiply both sides by 3x:
Distribute and simplify:
3 + 2 = 12x
Divide by 12:
5/12 = x
As this shows, sometimes solving equations requires a clever use of the Laws of Equality.
The Laws of Equality
Every equation is a balanced scale, and the Laws of Equality are simply the rules for “keeping the scale balanced,” that is, deducing other true equations. In a nutshell, the Laws of Equality say that
1. You may make changes to any equation, as long as you follow rules 2 and 3.
2. Whatever you do to one side of the equation, you must do to the other.
3. You may not perform undefined operations (like dividing by 0), or operations that have more than one possible result (like taking a square root).
If x2 = y2, then which of the following must be true?
I. x = y
III. x = |y|
B) I only
C) I and II only
D) I, II and III
(Medium-hard) This question tests your skills of deductive logic. Notice it is not asking which statements can be true, but rather which must be true. It seems that if we “unsquare” both sides of the original equation, we get the equation in I. If we divide the original equation by x on both sides, we get the equation in II. Does this mean that statements I and II are necessarily true? No, because we violated rule 3 in both cases. If x2 = y2, it does not follow that x = y. Notice that x could be 2 and y could be −2. These values certainly satisfy the original equation, but they do not satisfy the equations in I or III. They do, however, satisfy the equation in II, because 2 = (−2)2/2. However, statement II is still not necessarily true. What if x and y were both 0? This would satisfy the original equation, but 0 ≠ (0)2/(0) because 0/0 is undefined. Therefore, the correct answer is (A).
This example teaches us two lessons:
1. Before taking the square root of both sides of an equation, remember that every positive number has two square roots. For instance the square root of 9 is 3 or −3.
2. Before dividing both sides of an equation by an unknown, make sure it can’t equal 0.
Lesson 7: Making and analyzing graphs of linear equations
If m is a constant greater than 1, which of the following could be the graph in the xy-plane of x + my + m = 0?
(Medium-hard) First, we should try to get the equation into a more useful form. Let’s try the slope-intercept (y = mx + b) form:
x + my + m = 0
Subtract x and m:
my = −x − m
Divide by m:
This shows that the line has a slope of −1/m and a y-intercept of −1. Since the problem tells us that m is greater than 1, we know that the slope (−1/m) must be between −1 and 0. The only graph that satisfies these conditions is (B).
Thinking about slopes
It’s helpful to think of slope as the amount a line goes up (or down) for each step it takes to the right. Lines with a positive slope slant upward to the right, lines with a negative slope slant downward to the right, and lines with a 0 slope are horizontal. For instance, a line with slope −3 moves down 3 units for every unit step to the right.
Parallel and perpendicular lines
• Parallel lines have equal slopes.
• Perpendicular lines have slopes that are opposite reciprocals of each other. That is, if one line has a slope of a/b, its perpendicular has a slope of −b/a.
The points A(10, 4) and B(−2, k) are 13 units apart. Which of the following equations could describe the line that contains points A and B?
A) 13x + 12y = 178
B) 5x + 12y = 98
C) 5x − 12y = 98
D) 5x − 13y = −2
(Hard) Drawing a diagram will help us analyze this problem. Although we don’t know precisely where point B is, we know it is somewhere on the line x = −2. This gives us the following picture:
Next, notice that all of the equations given in the choices are in “standard” form, and in standard form the slope of the line is −a/b. Therefore, the slopes of these lines are, respectively, (A) −13/12, (B) −5/12, (C) 5/12, and (D) 5/13. Therefore, finding the slope of the line should help us choose the correct equation. Looking at the diagram more closely, notice that it includes two right triangles, and we can find the missing side of each one using the Pythagorean Theorem, or just by noticing that they are both 5-12-13 right triangles (52 + 122 = 132). Putting this information into the diagram shows us that B can therefore be at (−2, 9) or (−2, −1).
Therefore, the slope (rise/run) of the line containing B1 is −5/12, and the slope of the line containing B2 is 5/12. This means that our answer is either (B) or (C). How do we choose between them? Just remember that the line must contain the point (10, 4). If you plug x = 10 and y = 4 into these equations, only (B) works: 5(10) + 4(12) = 98.
Checking your work
Always check that your solutions satisfy your equations by plugging them back into the equations to verify.
Exercise Set 3 (No Calculator)
If x − 2(1 − x) = 5, what is the value of x?
If f(x) = −2x + 8, and f(k) = −10, what is the value of k?
What is the slope of the line that contains the points (−2, 3) and (4, 5)?
What is the slope of the line described by the equation ?
Line l is perpendicular to the line described by the equation 5x + 11y = 16. What is the slope of line l?
If , what is the value of x?
What is the y-intercept of the line containing the points (3, 7) and (6, 3)?
In the xy-plane, the graph of y = h(x) is a line with slope −2. If h(3) = 1 and h(b) = −9, what is the value of b?
If a train maintains a constant speed of 60 miles per hour, it can travel 4 miles per gallon of diesel fuel. If this train begins a trip with a full 200 gallon tank of diesel fuel, and maintains a speed of 60 miles per hour, which of the following equations represents the number of gallons, g, left in the tank t hours into the trip?
C) g = 200 −15t
The points A(2, 3) and B(m, 11), are 10 units apart. Which of the following equations could describe the line that contains points A and B?
A) 8x + 6y = 11
B) 8x − 6y = −2
C) 6x + 8y = 36
D) 6x − 8y = −12
The figure above shows a right triangle with vertices at the origin, (5, 6) and (k, 0). What is the value of k?
Exercise Set 3 (Calculator)
If the points (2, 4), (5, k), and (8, 20) are on the same line, what is the value of k?
Line l has a slope of 3 and a y-intercept of −4. What is its x-intercept?
If f(−1) = 1 and f(3) = 2 and f is a linear function, what is the slope of the graph y = f(x)?
If f(−1) = 1 and f(3) = 2 and f is a linear function, what is f(5)?
In the xy-plane, the graph of line n has an x-intercept of 2b and an y-intercept of −8b, where b ≠ 0. What is the slope of line n?
If , what is the value of x?
If the line 3x − 2y = 12 is graphed in the xy-plane, what is its x-intercept?
If the graphs of the equations 5x − 2y = 5 and 6x + ky = 9 are perpendicular, what is the value of k?
The net profit for the sales of a product is equal to the total revenue from the sales of that product minus the total cost for the sales of that product. If a particular model of calculator sells for $98, and the cost for making and selling n of these calculators is $(35n + 120,000), which of the following equations expresses the net profit in dollars, P, for making and selling n of these calculators?
A) P = 63n − 120,000
B) P = 63n + 120,000
C) P = 63(n − 120,000)
D) P = 63(n + 120,000)
Which of the following represents the equation of the line with an x-intercept of 5 and a y-intercept of 6?
The table above shows several ordered pairs corresponding to the linear function f. What is the value of a + b?
D) It cannot be determined from the information given.
EXERCISE SET 3 ANSWER KEY
Part 1: No Calculator
1. 7/3 or 2.33
x − 2(1 − x) = 5
x − 2 + 2x = 5
3x − 2 = 5
3x = 7
Divide by 3:
x = 7/3
f(k) = −2k + 8 = −10
−2k = −18
Divide by −2:
k = 9
3. 1/3 or .333
4. 10/3 or 3.33
Multiply by 2xy:
2y + y = 10x
3y = 10x
Divide by 3:
5. 11/5 or 2.2 The slope of the given line is −5/11, so the slope of the line perpendicular to it is 11/5.
6. 9/5 or 1.8
Multiply by 10:
(x + 1) + 4x = 10
5x + 1 = 10
5x = 9
Divide by 5:
x = 9/5
7. 11 There are a variety of ways of solving this problem, but perhaps the simplest is to draw a quick sketch:
Notice that to get from (6, 3) to (3, 7) we must go left 3 units and up 4 units (in other words, the slope is −4/3). If we simply repeat this from (3, 7), we arrive at the y-intercept, which is (0, 11).
8. 8 This line has a slope of −2 and contains the points (3, 1) and (b, −9). Therefore
Multiply by b − 3:
−2b + 6 = −10
−2b = −16
Divide by −2:
b = 8
9. C Since the tanks starts with 200 gallons, the amount it has left is 200 − the number of gallons used. The number of gallons used is
10. B Once again, a quick sketch can be very helpful. Notice that traveling from point
A(2, 3) to point B(m, 11) requires going up 8 units and right (or left) some unknown distance b. We can find b with the
82 + b2 = 102
64 + b2 = 100
b2 = 36
Take the square root:
b = 6
Therefore, m is either 2 − 6 = −4 or 2 + 6 = 8, and the slope of this line is either 8/6 = 4/3 or 8/(−6) = −4/3. The only equation among the choices that is satisfied by the ordered pair (2, 3) and has a slope of either 4/3 or −4/3 is (B).
11. D Recall that the slopes of perpendicular lines are opposite reciprocals. The slope of the segment from (0, 0) to (5, 6) is 6/5, so the slope of its perpendicular is −5/6.
−36 = 5(5 − k)
−36 = 25 − 5k
−61 = −5k
Divide by −5:
61/5 = k
Part 2: Calculator
12. 12 The slope of this line is ,
3k − 12 = 24
3k = 36
Divide by 3:
k = 12
13. 4/3 or 1.33 Since the slope and y-intercept are given, it is easy to express the linear equation in slope-intercept form: y = 3x − 4.
The x-intercept is the value of x on the line for which y = 0:
0 = 3x − 4
4 = 3x
Divide by 3:
4/3 = x
14. ¼ or .25 The line contains the points (−1, 1) and (3, 2), so its slope is
15. 5/2 or 2.5 Although we could solve this problem by deriving the linear equation, it is perhaps easier to take advantage of the result from question 14. The slope of 1/4 means that the y-coordinate of any point on the line increases by 1/3 each time the x-coordinate increases by 1. Since the x-coordinate increases by 2 between f(3) and f(5), the y-coordinate must therefore increase by 2(1/4) = 1/2, so f(5) = 2 + ½ = 2.5.
16. 4 The line contains the points (2b, 0) and (0, −8b); therefore, it has a slope of .
17. 3/5 or .6
Multiply by 5x:
10 + 2 = 20x
12 = 20x
Divide by 20:
x = 12/20 = 3/5
18. 4 The x-intercept is the value of x for which y = 0:
3x − 2(0) = 12
3x = 12
Divide by 3:
x = 4
19. 15 Recall that the slope of a line in standard form ax + by = c is −a/b. Therefore, the slope of 5x − 2y = 5 is 5/2 and the slope of 6x + ky = 9 is −6/k. If these lines are perpendicular, then their slopes are opposite reciprocals:
Multiply by 6:
k = 30/2 = 15
20. A The total revenue for selling n calculators at $98 each is $98n the cost for making and selling n calculators is $(35n + 120,000). Therefore the profit is $(98n − 35n − 120,000) = 63n − 120,000 dollars.
21. C This line contains the points (5, 0) and (0, 6) and therefore has a slope of .
Since its y-intercept is 6, its slope-intercept form is or, subtracting 6 from both sides,
22. B Since f is a linear function, it has a slope that we can call m. Recall that it’s often useful to think of the slope of a line as the “unit change,” that is, the amount that y changes each time x increases by 1. Since the x values increase by 1 with each step in our table, the y values must therefore increase by m with each step. This means that a = 8 − m and b = 8 + m. Therefore, a + b = 8 − m + 8 + m = 16.