PSAT/NMSQT Prep 2020 - Princeton Review 2020

Linear equations and graphs
The heart of algebra

Learning Objectives

After completing this chapter, you will be able to:

· Isolate a variable

· Translate word problems into equations

· Calculate the slope of a line given two points

· Write the equation of a line in slope-intercept form

· Discern whether the slope of a line is positive, negative, zero, or undefined based on its graph

· Describe the slopes of parallel and perpendicular lines

110/600 SmartPoints^®

How Much Do You Know?

Directions

Try the questions that follow. Show your work so that you can compare your solutions to the ones found in the Check Your Work section immediately after this question set. The “Category” heading in the explanation for each question gives the title of the lesson that covers how to solve it. If you answered the question(s) for a given lesson correctly, and if your scratchwork looks like ours, you may be able to move quickly through that lesson. If you answered incorrectly or used a different approach, you may want to take your time on that lesson.

(3x + 17) = (8x — 10)

1. Which value of x satisfies the equation above?

1. −61

2. −55

3. −41

4. −35

2. Jenna is renting a car while on a business trip. The cost is $54.95 per day, which is taxed at a rate of 6%. The car rental company charges an additional one-time, untaxed environmental impact fee of $10. Which of the following equations represents Jenna’s total cost, in dollars, for renting the car for d days?

1. c = (54.95 + 0.06d) + 10

2. c = 1.06(54.95d) + 10

3. c = 1.06(54.95d + 10)

4. c = 1.06(54.95 + 10)d

3. What was the initial amount of water in a barrel, in liters, if x liters remain after y liters were spilled and 6 liters were added?

1. x − y + 6

2. y − x + 6

3. x + y + 6

4. y + x − 6

4. Which of the following equations best describes the relationship shown in the table, where n indicates the number of cans sold and p represents the price in dollars of one can?

1. n = −20,000p + 25,000

2. n = −200p + 250

3. n = 200p + 250

4. n = 20,000p + 25,000

5. A freight company charges a flat insurance fee to deliver a package anywhere in the continental United States, plus an additional charge for each pound the package weighs. The graph shows the relationship between the weight of the package and the total cost to ship it. Based on the graph, how much would it cost to ship a 25-pound box?

1. $37.00

2. $48.00

3. $62.50

4. $74.50

6.

Check Your Work

1. A

Difficulty: Easy

Category: Solving Equations

Getting to the Answer: Look for a way to make the algebra manipulations easier (and quicker). Begin by multiplying both sides by 6 to eliminate the fractions. So, 3(3x + 17) = 8x − 10. Then, distribute to get 9x + 51 = 8x − 10. Combining like terms yields 9x − 8x = −10 − 51, and solving for x reveals that x = −61, which is (A).

2. B

Difficulty: Medium

Category: Word Problems

Getting to the Answer: Use the information in the question to write your own equation, then look for the answer choice that matches. Simplify your equation only if you don’t find a match. Start with the cost, not including tax or the environmental impact fee. If Jenna rents a car for d days at a daily rate of $54.95, the untaxed total is 54.95d. There is a 6% tax added to this amount, so multiply by 1.06 to get 1.06(54.95d). The $10.00 environmental impact fee is not taxed, so simply add 10 to your expression. The total cost is c = 1.06(54.95d) + 10, which matches (B), so you do not need to simplify.

3. D

Difficulty: Medium

Category: Word Problems

Getting to the Answer: Write an equation in words first, then translate from English to math. Finally, rearrange your equation to find what you’re interested in, which is the initial amount. Call the initial amount A. After you’ve written your equation, solve for A.

Amount now (x) = initial amount (A) minus y plus 6:

This is the same as y + x − 6, so (D) is correct.

You could also pick numbers to answer this question.

4. A

Difficulty: Medium

Category: Linear Graphs

Getting to the Answer: The answer choices are given in slope-intercept form, so start by finding the slope. To do this, substitute two pairs of values from the table into the slope formula, . Keep in mind that the projected number of cans sold depends on the price, so the price is the independent variable (x) and the projected number is the dependent variable (y). Using the points (0.75, 10,000) and (1.0, 5,000), the slope is:

This means that (A) must be correct because it is the only one that has a slope of —20,000. Don’t let (D) fool you—the projected number of cans sold goes down as the price goes up, so there is an inverse relationship, and the slope must be negative.

5. D

Difficulty: Hard

Category: Linear Graphs

Getting to the Answer: In a real-world scenario, the y-intercept of a graph usually represents a flat fee or an initial value. The slope of the line represents a unit rate, such as the cost per pound. The y-intercept of the graph is 12, so the flat fee is $12. To find the cost per pound (the unit rate), substitute two points from the graph into the slope formula. Using the points (0, 12) and (2, 17), the unit rate is , which means $2.50 per pound will be added to the cost. The total cost to ship a 25-pound box is $12 + 2.50(25) = $12 + $62.50 = $74.50, which is (D).

Solving Equations

Learning Objective

After this lesson, you will be able to:

· Isolate a variable

To answer a question like this:

1.

Which of the following values is equal to x ?

1. −36

2. 4

3. 12

4. 20

You need to know this:

Isolating a variable means getting that variable by itself on one side of the equation. To do this, use inverse operations to manipulate the equation, remembering that whatever you do to one side of the equation, you must do to both sides.

You need to do this:

It usually makes sense to proceed in this order:

1. Eliminate any fractions.

2. Collect and combine like terms.

3. Divide to leave the desired variable by itself.

Explanation:

Eliminate the fractions by multiplying both sides of the equation by 6:

In order to collect all the x terms on one side, you’ll first need to distribute the 2 on the left side of the equation:

Next, subtract 5x from both sides:

Finally, subtract 24 from both sides:

Therefore, (A) is correct.

Try on Your Own

Directions

Take as much time as you need on these questions. Work carefully and methodically. There will be an opportunity for timed practice at the end of the chapter.

1. HINT: Since the answer choices in Q1 involve decimals, convert the fractions to decimals.

2. Which approximate value of x satisfies the equation shown?

1. 4.3

2. 4.6

3. 6.6

4. 6.8

3. HINT: Is there a simpler way to solve Q2 than finding the value for n?

4. If , then what does equal?

5. In the equation shown, what is the value of n ?

1.

2.

3.

4.

6. If 36 + 3(4x − 9) = c(2x + 1) + 25 has no solution and c is a constant, what is the value of c ?

1. −3

2. 3

3. 6

4. 12

7. HINT: There’s no need to try to isolate the variable the equation for Q5.

8. In the equation , which of the following is a possible value of x — 4 ?

1.

2. 4

3.

4. 5

9.

Word Problems

Learning Objective

After this lesson, you will be able to:

· Translate word problems into equations

To answer a question like this:

1. A local gym sells two types of memberships. One package costs $325 for one year of membership with an unlimited number of visits. The second package has a $185 enrollment fee, includes five free visits, and costs an additional $4 per visit after the first five. How many visits over a one-year period would a person who purchases the second package need to make for the cost to be the same as the one-year unlimited membership?

1. 30

2. 35

3. 40

4. 45

You need to know this:

The PSAT likes to test your understanding of how to describe real-world situations using math equations. For some questions, it will be up to you to extract and solve an equation; for others, you’ll have to interpret an equation in a real-life context. The following table shows some of the most common phrases and mathematical equivalents you’re likely to see on the PSAT.

You need to do this:

When translating from English to math, start by defining the variables, choosing letters that make sense. Then, break the question down into small pieces, writing down the translation for one phrase at a time.

Explanation:

The phrase “how many visits” indicates an unknown, so you need a variable. Use an intuitive letter to represent the number of visits; call it v. The question asks when the two memberships will cost the “same amount,” so write an equation that sets the total membership costs equal to each other.

The first membership type costs $325 for unlimited visits, so write 325 on one side of the equal sign. The second type costs $4 per visit (not including the first 5 visits), or 4(v − 5), plus a flat $185 enrollment fee, so write 4(v − 5) + 185 on the other side of the equal sign. That’s it! Now, solve for v:

The answer is (C).

Try on Your Own

Directions

Take as much time as you need on these questions. Work carefully and methodically. There will be an opportunity for timed practice at the end of the chapter.

1. An automotive insurer computes the charge for an insurance policy by starting with a fixed fee for administrative expenses and then adding a percentage of the value of the car, v. If the total price for an automobile policy is given by the function T = 0.02v + 25, then the value 0.02 best represents which of the following?

1. The fee for the administrative expenses

2. The value of the car

3. The total price of the policy

4. The percentage of the value of the car

2. Ibrahim has a contract for a cell phone plan that includes the following rates: the plan has a fixed cost of $50 per month, a data plan that provides 2 gigabytes (GB) of data for free and $8 for each GB of data after that, and a text message plan that costs $0.10 per text message sent. Which of the following equations represents the amount of money, in dollars, that Ibrahim will spend as long as he uses at least 2 GB of data? (Assume d = dollars, g = number of GB of data used, and t = number of text messages sent.)

1. d = 50 + 8g + 0.1t

2. d = 50 + (8g — 2) + 0.1t

3. d = 50 + 8(g — 2) + 0.1t

4. d = 5,000 + 800g + 10t

3. HINT: For Q8, take the time to determine which number goes with which variable before attempting to build the entire equation.

4. An online video game service has two different packages available. The first is an Unlimited Rental Package that costs $250 for 3 months and allows customers to rent as many games as they want. The second is the Standard Rental Package that initially has a flat fee of $130 but costs $4 per rental in the 3-month period. How many rentals on the Standard Rental Package would it take for both packages to equal the same price over 3 months?

1. 2

2. 30

3. 96

4. 120

5. A produce stand normally sells watermelons for $0.60 per pound. On Mondays, it sells watermelons at a 20% discount. The stand also sells sweet potatoes for $0.79 each. Which of the following represents the total cost, c, if a customer buys 4 sweet potatoes and a watermelon weighing p pounds on a Monday?

1. c = 0.2p + 0.79

2. c = 0.48p + 3.16

3. c = 0.6p + 0.79

4. c = 0.6p + 3.16

6. HINT: For Q10, if a arrows hit the inner circle, how many hit the outer circle in terms of a?

7. Juan is on his school’s archery team. In a match, he gets 12 arrows to shoot at a target. He gets 8 points if the arrow hits the inner circle of the target and 4 points if it hits the outer circle. Which of the following equations represents Juan’s total score if a of his arrows hit the inner circle and the rest hit the outer circle?

1. p = 8a

2. p = 8a + 4

3. p = 48 + 4a

4. p = 96 − 4a

8.

Linear Graphs

Learning Objective

After this lesson, you will be able to:

· Calculate the slope of a line given two points

· Write the equation of a line in slope-intercept form

To answer a question like this:

1. Which of the following equations describes the line that passes through the points (2, —4) and (—2, 2) ?

1.

2.

3.

4.

You need to know this:

The answer choices in this question are written in slope-intercept form: y = mx + b. In this form of a linear equation, m represents the slope of the line and b represents the y-intercept. You can think of the slope of a line as how steep it is. The y-intercept is the point at which the line crosses the y-axis and can be written as the ordered pair (0, y).

You can calculate the slope of a line if you know any two points on the line. The formula is , where (x₁, y₁) and (x₂, y₂) are the coordinates of the two points on the line.

A line that moves from the bottom left to the top right has a positive slope. A line that moves from the top left to the bottom right has a negative slope. A horizontal line has a slope of zero and a vertical line has an undefined slope.

Some PSAT questions ask about parallel or perpendicular lines. Parallel lines have the same slope, while perpendicular lines have negative reciprocal slopes.

You need to do this:

· Find the slope of the line.

· Write the equation in slope-intercept form, substituting the value of the slope you found and one of the known points for x and y.

· Solve for the y-intercept.

Explanation:

In the question above, . Of the answer choices, only (C) and (D) have negative slopes, so rule out (A) and (B).

To find the y-intercept of the line, write the equation for the line in slope-intercept form and plug in one of the known points for x and y:

Therefore, (D) is correct.

Another way to solve this question would be to plot the two points, graph the line, check where it crosses the y-axis to find the y-intercept, and count how many spaces the line moves up for each space it moves to the right to find the slope.

Try on Your Own

Directions

Take as much time as you need on these questions. Work carefully and methodically. There will be an opportunity for timed practice at the end of the chapter.

1. HINT: For Q11, remember that parallel lines have the same slope and perpendicular lines have opposite sign reciprocal slopes.

2. A line, t, is perpendicular to and passes through the point (3, 5). What is the equation of line t ?

1.

2.

3.

4.

3. HINT: Convert the word problem in Q12 to an equation in slope-intercept form, y = mx + b.

4. In June, Marie notices that the price of gas is increasing. She records the price of gas 4, 8, and 10 months after her initial observation. If y is the price of gas, in dollars per gallon, and x is the number of months after June, which linear equation represents the correct relationship between y and x ?

1. y = 0.33x + 0.67

2. y = 0.67x + 1.33

3. y = 1.33x + 2

4. y = 2.67x + 5.67

5. HINT: For Q13, focus on where the answer choices differ in their equations: the y-intercept.

6. The graph above shows the volume of a sample of gas as it is cooled. If T is the temperature of the gas in degrees Celsius and V is the volume in liters, which of the following equations, when plotted, could produce the graph shown?

1. V = 0.004T + 100

2. V = 0.004T

3. V = 0.004T + 1

4. V = 0.004T − 0.25

7. A driving range sells golf balls in different quantities. The figure shows the costs of the various quantities. According to the figure, what is the cost of a single golf ball?

1. $0.60

2. $1.67

3. $3.00

4. $5.00

8. For what value of y does the graph of cross the y-axis?

9.

On Test Day

Remember that the PSAT doesn’t ask you to show your work. If you find the algebra in a question challenging, there is often another way to get to the answer.

Try this question first using algebra and then using the Picking Numbers strategy you learned in chapter 3. Time yourself. Which approach do you find easier? Which one was faster? Did you get the correct answer both times? Remember your preferred approach and try it first if you see a question like this on test day.

1. Which of the following statements is NOT true if the above equation is true?

1.

2.

3.

4.

The answer and explanation can be found at the end of this chapter.

How Much Have You Learned?

Directions

For testlike practice, give yourself 15 minutes to complete this question set. Be sure to study the explanations, even for questions you got right. They can be found at the end of this chapter.

1. Line m is parallel to line k and passes through (5, 5). The slope of line k is and line k passes through the origin. Which of the following is the equation for line m ?

1.

2.

3.

4.

2. Which value of x makes the equation true?

1. 7

2. 9

3. 17

4. 19

3. What is the value of z in the equation above?

1.

2.

3.

4.

4. Sandy works at a tire store. She gets paid $70 for a day’s work, plus a commission of $14 for each tire she sells. Which of the following equations represents the relationship between one day of Sandy’s pay, y, and the number of tires she sells, x ?

1. x = 14y + 70

2. x = 70y + 14

3. y = 14x + 70

4. y = 70x + 14

5. The figure above shows the cost of going to a certain carnival. What does the y-intercept most likely represent?

1. A flat entrance fee

2. The cost of riding 8 rides

3. The cost of attending the carnival 8 times

4. The total cost of attending the carnival and riding 1 ride

6. For a car, depreciation is a reduction in value due to use or age. The value of a new car starts to depreciate as soon as the car is purchased and driven for the first time. If the equation y = −0.15x + 27,000 represents the estimated value of a certain car, taking into account depreciation over time, what does 27,000 most likely represent?

1. The depreciation rate

2. The current value of the car

3. The original purchase price of the car

4. The value of the car after 0.15 years of ownership

7. What is the value of h in the equation above?

1. h = —5.5

2. h = 5.5

3. There is no value of h for which the equation is true.

4. There are infinitely many values of h for which the equation is true.

8. What value of y satisfies the equation above?

9. A cybercafe is a place that provides Internet access to the public, usually for a fee, along with snacks and drinks. Suppose a cybercafe charges a base rate of $25 to join for a year, an additional $0.30 per visit for the first 50 visits, and $0.10 for every visit after that. How much does the cybercafe charge for a year in which 72 visits are made?

1. $32.20

2. $36.60

3. $42.20

4. $46.60

10.An Internet provider charges k dollars for the first hour of use in a month and m dollars per hour for every additional hour used that month. If Jared paid $65.50 for his Internet use in one month, which of the following expressions represents the number of hours he used the Internet that month?

1.

2.

3.

4.

11.

Reflect

Directions: Take a few minutes to recall what you’ve learned and what you’ve been practicing in this chapter. Consider the following questions, jot down your best answer for each one, and then compare your reflections to the expert responses on the following page. Use your level of confidence to determine what to do next.

What should you do to isolate a particular variable in an equation?

What types of keywords should you look for when translating English into math?

What is the most useful equation for a line in the coordinate plane? Why?

When the PSAT gives you two points on a line, what can you figure out?

How are parallel and perpendicular lines related to each other?

EXPERT RESPONSES

What should you do to isolate a particular variable in an equation?

Perform inverse operations until the variable is by itself on one side of the equal sign. If the equation has fractions, make them disappear by multiplying both sides of the equation by the denominator(s). If like terms appear on different sides of the equation, collect them on the same side so that you can combine them.

What types of keywords should you look for when translating English into math?

Look for keywords that signal equality (“is,” “has,” “was”), variable names (“Marina’s age,” “the cost of one bathtub”), or one of the four arithmetic operations (addition, subtraction, multiplication, and division).

What is the most useful equation for a line in the coordinate plane? Why?

The best equation is slope-intercept form, y = mx + b, because it tells you the slope (m) and the y-intercept (b). Conversely, if you need to derive an equation yourself, you can plug the slope and y-intercept into slope-intercept form and you’re done.

When the PSAT gives you two points on a line, what can you figure out?

If you know two points, you can figure out the slope of the line with the equation . From there, you can plug one of the points and the slope into slope-intercept form and find the y‑intercept.

How are parallel and perpendicular lines related to each other?

Parallel lines never intersect and they have equal slopes. Perpendicular lines intersect at a 90° angle and they have negative reciprocal slopes.

NEXT STEPS

If you answered most questions correctly in the “How Much Have You Learned?” section, and if your responses to the Reflect questions were similar to those of the PSAT expert, then consider Linear Equations and Graphs an area of strength and move on to the next chapter. Come back to this topic periodically to prevent yourself from getting rusty.

If you don’t yet feel confident, review those parts of this chapter that you have not yet mastered. In particular, review the variable isolation example in the Solving Equations lesson and the definition of slope-intercept form in the Linear Graphs lesson. Then, try the questions you missed again. As always, be sure to review the explanations closely.

Answers and Explanations

1. D

Difficulty: Medium

Getting to the Answer: Multiply both sides by 7 to eliminate the fraction outside the parentheses. Then, because the choices are decimals, convert all fractions to decimal form: 8(x − 0.46) + 28(x + 0.89) = 266. Distribute to get 8x − 3.68 + 28x + 24.92 = 266, then combine like terms to yield 36x = 266 + 3.68 − 24.92, or 36x = 244.76. Divide both sides by 36 to find that x = 6.8, which is (D).

2. 1/3 or .333

Difficulty: Medium

Getting to the Answer: Notice that n − 2 is in both the equation and the desired expression, so isolate n − 2 and plug it into the expression. Determine the value of n − 2 by dividing both sides of 3(n − 2) = 6 by 3. Now you know that n − 2 = 2, so plug 2 into the numerator of the desired expression. To find the value of the denominator, add 2 to both sides of the given equation to find that n = 4. Plug 4 into n + 2, which leaves . Grid in 1/3 or .333 and move on to the next question.

Note that you could also solve the equation for n and substitute the result into the desired expression, but this may take a bit more time.

3. A

Difficulty: Medium

Getting to the Answer: Start by simplifying the numerators. Don’t forget to distribute the negative to both terms inside the parentheses on the right side of the equation:

Next, cross-multiply and solve for n using inverse operations:

This matches (A).

4. C

Difficulty: Medium

Getting to the Answer: If a linear equation has no solution, the variables cancel out, leaving two numbers that are not equal to each other. Start by simplifying the left side of the equation:

The variable term on the left is 12x. Because the variable terms must cancel, the right side of the equation must also have a 12x, so it must be that c = 6, which is (C).

5. A

Difficulty: Medium

Strategic Advice: When a question asks for a value of an expression, isolate that expression on one side of the equation. Do not solve for the variable because you’ll have the extra step of plugging that value into the expression.

Getting to the Answer: Notice that the question asks for a possible value of x − 4, not a possible value of x. Isolate x − 4 on one side of the equation by multiplying both sides by x − 4 and taking the square root of both sides:

Thus, (A) is correct. Watch out for traps like (C), which is a possible value of x.

An alternative strategy is to backsolve, plugging each choice in for x — 4 to see which results in a true statement.

1. D

Difficulty: Easy

Getting to the Answer: The total insurance bill consists of a flat fee and a percentage of the value of the car. The administrative fee does not depend on the value of the car and therefore should not be multiplied by v. This means that 25 is the administrative fee. The other expression in the equation, 0.02v, represents the percentage of the value of the car times the car’s value (which the question tells you is v). Therefore, 0.02 must represent the value of the car as a percentage, which is (D).

2. C

Difficulty: Medium

Getting to the Answer: You know there will be a flat fee of $50, and for text messages you’ll be looking for 0.1t. This eliminates (D). If you’re stuck on the data plan cost, plug in some numbers. For g = 2, you wouldn’t expect there to be an additional fee for data usage. For g = 3, you’d expect to see an $8 charge, and for g = 4, you’d expect to see a $16 charge. The only choice that reflects this is (C).

3. B

Difficulty: Medium

Getting to the Answer: Translate from English into math by assembling equations and setting them equal to each other. Let u = Unlimited Rental Package, s = Standard Rental Package, and r = number of rentals. Using the information about each package, you know that u = 250 and s = 130 + 4r. Now, solve for r when u = s:

Choice (B) is correct.

4. B

Difficulty: Medium

Getting to the Answer: Write the equation in words first, then translate from English to math. The total cost, c, is the weight of the watermelon in pounds, p, multiplied by the sale price. Since the purchase is made on Monday, the sale price is: $0.60 × 80% = 0.6 × 0.8 = 0.48. This gives the cost of the watermelons: 0.48p. Now, add the cost of four sweet potatoes, 0.79 × 4 = 3.16, to get the equation c = 0.48p + 3.16, which matches (B).

You could also use the Picking Numbers strategy: pick a number for the weight of the watermelon and calculate how much it would cost (on sale). Next, add the cost of four sweet potatoes. Finally, find the equation that gives the same amount.

5. C

Difficulty: Hard

Getting to the Answer: The key to answering this question is to determine how many arrows hit each circle. If there are 12 arrows total and a hit the inner circle, the rest, or 12 — a, must hit the outer circle. Now, write the expression in words: points per inner circle, 8, times number of arrows in inner circle, a, plus points per outer circle, 4, times number of arrows in outer circle, 12 — a. Next, translate the words into numbers, variables, and operations: 8a + 4(12 — a). This is not one of the answer choices, so simplify the expression by distributing the 4 and then combining like terms: 8a + 4(12 — a) = 8a + 48 — 4a = 4a + 48, so the equation is p = 4a + 48. Rearrange the order of the terms on the right side to arrive at (C).

1. A

Difficulty: Medium

Getting to the Answer: The first useful piece of information is that the slope of the line perpendicular to line t is . Perpendicular lines have negative reciprocal slopes, so the slope of line t is . Eliminate (B) and (D) because they have the incorrect slopes.

Plug the values for the slope and the coordinate point (3, 5) into the slope-intercept equation to solve for b:

Eliminate (C) because it does not have the correct y-intercept. Choice (A) is correct.

2. A

Difficulty: Medium

Getting to the Answer: The question tells you that the relationship is linear, so start by finding the the slope, m, using any two pairs of values from the table and the slope formula. Next, substitute the slope and any pair of values from the table, such as (4, 2) and (10, 4), into the equation y = mx + b and solve for b. Finally, use the values of m and b to write the function:

You can stop right there! Only (A) has a slope of 0.33, so it must be the correct answer. For the record:

3. C

Difficulty: Medium

Getting to the Answer: Temperature, T, is the independent variable on the x-axis, and volume, V, is the dependent variable on the y-axis. Therefore, the skeleton of the equation you’re looking for is V = mT + b. Because each answer choice has a different y-intercept, finding b is enough to get the right answer; there is no need to determine the slope. Although the graph is not centered around the origin, you can still find the y-intercept. In this case, it’s 1. This eliminates every answer choice except (C), which is correct.

4. B

Difficulty: Medium

Getting to the Answer: The x-axis represents the number of golf balls, so find 1 on the x-axis and trace up to where it meets the graph of the line. The y-value is somewhere between $1 and $2, so the only possible correct answer is $1.67, which is (B).

You could also find the unit rate by calculating the slope of the line using two of the points shown on the graph. The graph rises 5 units and runs 3 units from one point to the next, so the slope is , or 1.67.

5. 24

Difficulty: Medium

Getting to the Answer: The place where the line crosses the y-axis is the y-intercept, or b when the equation is written in slope-intercept form (y = mx + b), so rewrite the equation in this form:

The y-intercept is 24.

Because the y-intercept of a graph is always of the form (0, y), you could also substitute 0 for x in the original equation and solve for y.

1. B

Difficulty: Hard

Category: Solving Equations

Strategic Advice: Watch out for the word NOT in the question stem.

Getting to the Answer: To find the answer using Picking Numbers, take advantage of the fact that the equation is a proportion (that is, two fractions equal to each other). If b = 5 and 2(a − 4) = 4, then both fractions will be the same and the numbers you’ve picked will be valid. Solve for a: a − 4 = 2, so a = 6. Now, plug b = 5 and a = 6 into the choices, looking for the one that is NOT true:

(A) , eliminate.

(B)

You’re done: pick (B) and move on. For the record:

(C) , eliminate.

(D) , eliminate.

To solve this question using algebra, first cross-multiply and simplify the original equation:

Then, repeat this process for each answer choice, looking for the one that does NOT yield the same equation:

(A) , eliminate.

(B)

This equation is different from the one in the question, so (B) is correct. For the record:

(C) , eliminate.

(D) , eliminate.

1. D

Difficulty: Medium

Category: Linear Graphs

Getting to the Answer: This question provides an equation for a line parallel to line m and a coordinate point for line m, (5, 5). The slope of the parallel line, which is , is the same as the slope of line m. Eliminate (B) and (C) because they have the incorrect slope.

Plug the values for the slope and the coordinate point (5, 5) into the slope-intercept equation to solve for b:

Eliminate (A) because it does not have the correct y-intercept. Only (D) is left and is correct. To confirm, plug the values for the slope and the y-intercept into the slope-intercept equation to get .

2. D

Difficulty: Easy

Category: Solving Equations

Getting to the Answer: Distributing the will result in messy calculations, so clear the fraction instead. Multiply both sides of the equation by the reciprocal of , which is , and isolate x:

Choice (D) is correct.

Alternatively, you could backsolve. Say you started with (B). The left side of the equation would become , which is smaller than 12, so you would move to (C) next. Plugging in 17 gives . This is still smaller than 12 since 12(3) = 36, so the correct answer again must be (D).

3. D

Difficulty: Medium

Category: Solving Equations

Getting to the Answer: Simplify the numerators as much as possible, then isolate the variable. Begin by combining like terms on both sides of the equation. Once complete, cross-multiply and solve for z:

Choice (D) is correct.

4. C

Difficulty: Easy

Category: Word Problems

Getting to the Answer: When writing a linear equation, a flat rate is a constant while a unit rate is always multiplied by the independent variable. For one day of work, Sandy is paid $70, which is a flat rate and should be the constant in the equation. You can identify the unit rate by looking for words like per or for each. The clue “for each” tells you to multiply $14 by the number of tires she sells, so the equation is pay = 14 × number of tires + 70, or y = 14x + 70. This matches (C).

5. A

Difficulty: Easy

Category: Linear Graphs

Getting to the Answer: Read the axis labels carefully. The y-intercept is the point at which x = 0, which means the number of rides is 0. The y-intercept is (0, 8). This means the cost is $8 before riding any rides, and therefore 8 most likely represents a flat entrance fee, (A).

6. C

Difficulty: Easy

Category: Word Problems

Getting to the Answer: Because the car’s value is always declining from the time it is purchased, its greatest value is its purchase price when new. The car’s price at any time can be represented as the purchase price minus some value that depends on the car’s age. In the given formula, 0.15x is subtracted from 27,000, suggesting that 0.15x represents the decline in value over time and 27,000 represents the car’s original purchase price, choice (C).

7. C

Difficulty: Medium

Category: Solving Equations

Getting to the Answer: Look for a way to make the math easier, such as clearing the fractions first. To do this, multiply both sides of the equation by 6, then solve for h using inverse operations:

Because the variable terms in the equation cancel out, and 15 does not equal 48, the equation has no solution. In other words, there is no value of h that satisfies the equation, so (C) is correct.

8. 14

Difficulty: Easy

Category: Solving Equations

Getting to the Answer: Eliminate the fractions to simplify the math. Multiply both sides of the equation by 4, then solve for y using inverse operations:

Grid in 14.

9. C

Difficulty: Easy

Category: Word Problems

Getting to the Answer: When a question involves several rates, break the situation into separate, manageable pieces and deal with each in turn. Visitors must pay the cyber cafe a flat $25 fee to join regardless of the number of visits. The first 50 visits cost $0.30 each, or 50($0.30) = $15. The remaining 72 — 50 = 22 visits cost $0.10 each, or 22($0.10) = $2.20. The total cost for 72 visits is therefore $25.00 + $15.00 + $2.20 = $42.20, which is (C).

10.D

Difficulty: Hard

Category: Word Problems

Getting to the Answer: Let h be the number of hours Jared spent using the Internet during the month. The first hour costs k dollars and the remaining hours (h − 1) are charged at the rate of m dollars per hour. Therefore, the total charge for a month is k + (h − 1)m. Set this equal to the amount Jared paid and solve for h. Note that you’re not going to get a numeric answer because the question doesn’t give you the actual rates:

This expression matches (D). Note that you could also use Picking Numbers to answer this question.