PSAT/NMSQT Prep 2022 - Eggert M.D., Strelka A. 2022

Reflect
Linear equations and graphs
The heart of algebra

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?

= mx + b The best equation is slope-intercept form, y , 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-interceptform 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.

6. 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).

7. 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).

8. 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.

9. 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.

10. 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).

11. 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.

12. 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:

13. 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 correct 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.

14. 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.

15. 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.

16. 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.

17. 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 .

18. 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).

19. A

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 (A) is correct.

20. 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).

21. 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).

22. 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).

23. 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.

24. 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.

25. 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).

26. 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.