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

Reflect
Systems of linear equations
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.

When is substitution a good choice for solving a system of equations?

When is combination a good choice for solving a system of equations?

What does it mean if a system of equations has no solution? Infinitely many solutions?

Expert Responses

When is substitution a good choice for solving a system of equations?

Substitution works best when at least one of the variables has a coefficient of 1, making the variable easy to isolate. This system, for example, is well suited for substitution:

a + 3b = 5

4a − 6b = 21

That’s because in the first equation, you can easily isolate the a as a = 5 − 3b and plug that in for a in the other equation. By contrast, substitution would not be a great choice for solving this system:

2a + 3b = 5

4a − 6b = 21

If you used substitution now, you’d have to work with fractions, which is messy.

When is combination a good choice for solving a system of equations?

Combination is always a good choice. It is at its most difficult in systems such as this one:

2a + 3b = 5

3a + 5b = 7

Neither a -coefficient is a multiple of the other, and neither b -coefficient is a multiple of the other, so to solve this system with combination you’d have to multiply both equations by a constant (e.g., multiply the first equation by 3 and the second equation by 2 to create a 6 a term in both equations). But substitution wouldn’t be stellar in this situation, either.

Note that combination may be particularly effective when the PSAT asks for a variable expression. For example, if a question based on the previous system of equations asked for the value of 5 a+8 b , then you could find the answer instantly by adding the equations together.

What does it mean if a system of equations has no solution? Infinitely many solutions?

A system of equations with no solution represents two parallel lines, which never cross. The coefficient of a variable in one equation will match the coefficient of the same variable in the other equation, but the constants will be different. For example, this system has no solution:

2x + 3y = 4

2x + 3y = 5

Subtracting one equation from the other yields the equation 0 = − 1, which makes no sense.

If a system of equations has infinitely many solutions, then the two equations represent the same line. For example, this system has infinitely many solutions:

2x + 3y = 4

4x + 6y = 8

Dividing the second equation by 2 yields 2x + 3y = 4, so while the two equations look different, they are actually the same.

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 Systems of Linear Equations 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 mechanics for solving a system of equations by substitution and by combination. Then, try the questions you missed again. As always, be sure to review the explanations closely.

Answers and Explanations

1. C

Difficulty: Easy

Getting to the Answer: Translate the words in the question into equations. Let n be the original number of nickels and d be the number of dimes. That there were “twice as many nickels as dimes” means that n = 2d. When 4 nickels are added, the number of nickels is 3 times the number of dimes. Thus, n + 4 = 3d. Substitute 2d for n in the second equation: 2d + 4 = 3d. Subtract 2d from each side to get 4 = d. The question asks for the original number of coins. The original number of nickels is n = 2d = 8 and the total number of coins is 4 + 8 = 12, which is (C).

2. 4/3 or 1.33

Difficulty: Medium

Getting to the Answer: Start by isolating c in the second equation: c = 5b − 4. Then, substitute into the first equation and solve:

Grid in 4/3 or 1.33 and move on.

3. C

Difficulty: Medium

Getting to the Answer: Because x has a coefficient of 1 in the second equation, solve the system using substitution. Before you select your answer, make sure you found the right quantity (the difference between x and y).

First, solve the second equation for x and substitute:

Next, substitute this value back into x = 4 + 3y and simplify:

Finally, subtract x − y to find the difference:

Hence,(C) is correct. While substitution is a valid way to solve this because the second equation readily gives you x in terms of y, you could have just restated the first equation as 5x − 5y = 10 and, therefore, x − y = 2.

4. C

Difficulty: Medium

Getting to the Answer: Since the question asks for r + 6, substitute by solving for s using the second equation, r + s = 12, so s = 12 − r. Substitute 12 − r into the first equation to get 3r + 2(12 − r) = 24. Distribute the 2 to get 3r + 24 − 2r = 24. Next, combine like terms: 3r − 2r = 24 − 24, which yields r = 0. Remember that the question asks for r + 6, not r by itself!

Choice (C) is correct.

5. D

Difficulty: Hard

Getting to the Answer: Create a system of two linear equations where t represents tables with 2 chairs and f represents tables with 4 chairs. The first equation should represent the total number of tables, each with 2 or 4 chairs, or t + f = 25. The second equation should represent the total number of chairs. Because t represents tables with 2 chairs and f represents tables with 4 chairs, the second equation should be 2t + 4f = 86. Now, solve the system using substitution. Solve the first equation for t in terms of f so that when you substitute the result into the second equation, you can solve directly for f:

There are 18 tables with 4 chairs each, (D). This is all the question asks for, so you don’t need to find the value of t.

6. A

Difficulty: Easy

Getting to the Answer: Quickly compare the two equations. The system is already set up perfectly to solve using combination, so add the two equations to cancel −4y and 4y. Then, solve the resulting equation for x. Remember, the question asks for the y-coordinate of the solution, so you will need to substitute x back into one of the original equations and solve for y:

Thus, (A) is correct.

7. D

Difficulty: Easy

Getting to the Answer: If you’re not asked to find the value of an individual variable, the question may lend itself to combination. This question asks for 3b − 2c, so don’t waste your time finding the variables individually if you can avoid it. After rearranging the equations so that variables and constants are aligned, you can add the equations together:

This matches (D).

8. B

Difficulty: Easy

Getting to the Answer: Looking at the coefficients of the two equations, you’ll notice that multiplying the second equation by −3 will allow you to eliminate the b terms:

Solving the resulting equation gives a = 4. Choice (B) is the only choice that contains this value for a, so it must be correct.

9. 51

Difficulty: Medium

Getting to the Answer: Rather than multiplying just one equation by a factor, you’ll need to multiply both by a factor to use combination. Suppose you want to eliminate x. The coefficients of the x terms are 2 and 5, so you need to multiply the equations by numbers that will give you −10 and 10 as your new x term coefficients. To do this, multiply the first equation by −5 and the second equation by 2:

−5(2x + 5y = 49)

2(5x + 3y = 94)

Add the resulting equations:

Solving for y gives you 3. Next, plug 3 back in for y in either equation and solve for x, which equals 17. Multiplying x and y together yields 51. Grid in 51.

10. B

Difficulty: Medium

Getting to the Answer: Translate English into math to extract what you need. First, define the variables using letters that make sense. Use c for children and a for adults. Now, break the word problem into shorter phrases: children’s tickets sold for $8 each; adult tickets sold for $12 each; 60 people attended the concert; $624 was collected in ticket money. Translating each phrase into a math expression will produce the components needed:

Children’s tickets (c) cost $8 each → 8c

Adult tickets (a) cost $12 each → 12a

60 people attended the concert → c +a = 60

$624 was collected in ticket money → Total $ = 624

Now, put the expressions together in a system:

You can solve for the variables using combination by multiplying the first equation by 8 and subtracting it from the second equation:

Plug this value into c + a = 60 to find that c = 24. Remember, the question asks for the difference between the number of adults and the number of children, so the correct answer is 36 − 24 = 12, which corresponds to (B).

11. C

Difficulty: Medium

Getting to the Answer: You are told that the solution to the system is x = −5 and y = 2. Substitute these values into both equations to find h and k:

So, , making (C) correct.

12. B

Difficulty: Hard

Getting to the Answer: A system of equations that has infinitely many solutions describes a single line. Therefore, manipulation of one equation will yield the other. Look at the constant terms: to turn the 24 into −2, divide the first equation by −12:

The y term and the constant in the first equation now match those in the second. All that’s left is to set the coefficients of x equal to each other: . Choice (B) is correct.

Note that you could also write each equation in slope-­intercept form and set the slopes equal to each other to solve for q.

13. A

Difficulty: Hard

Getting to the Answer: A system of linear equations that has no solution should describe two parallel lines. This means the coefficients of the variables should be the same (so the slopes of the lines are the same). Only the constant should be different (so the y-intercepts are not the same). The easiest way to make the coefficients the same is to manipulate the second equation. Multiplying the second equation by 40 would make the coefficients of x the same in both equations: 8x + 40 zy = 20. Now, equate the coefficients of y to get 4y = 40 zy. Solve for z to reveal that , which is (A). Alternatively, you could write each equation in slope-intercept form and set the slopes equal to each other to solve for z.

14. A

Difficulty: Hard

Getting to the Answer: One way to answer the question is to think about the graphs of the equations. Graphically, a system of linear equations that has no solution indicates two parallel lines or, in other words, two lines that have the same slope. Write each of the equations in slope-intercept form, y = mx + b, and set their slopes, m, equal to each other to solve for w.

First equation:

Second equation:

Set the slopes equal:

This matches (A). Alternatively, you could manipulate the first equation to make the x-coefficients the same and then equate the coefficients of y to solve for w.

15. B

Difficulty: Hard

Getting to the Answer: A system of linear equations has infinitely many solutions if both lines in the system have the same slope and the same y-intercept (in other words, they are the same line). Write each of the equations in slope-intercept form, y = mx + b. Their slopes should be the same. To find c, set the y-intercepts, b, equal to each other and solve. Before rewriting the equations, multiply the first equation by 6 to make it easier to manipulate.

First equation:

Second equation:

Set the y-intercepts equal:

Hence, (B) is correct.

16. C

Difficulty: Medium

Category: Combination

Strategic Advice: The numbers here are fairly large, so substitution is not likely to be convenient. Moreover, the y-coefficients have the same absolute value, so combination will likely be the faster way to solve.

Getting to the Answer: Start by writing the second equation in the same form as the first, then use combination to solve for x:

Thus, (C) is correct.

If you feel more comfortable using substitution, you can maximize efficiency by solving one equation for 7y and substituting that value into the other equation:

Note that the arithmetic is fundamentally the same, but the setup using combination is quicker and visually easier to follow.

17. D

Difficulty: Medium

Category: Combination

Strategic Advice: When a question asks for a sum or difference of variables, consider solving by combination.

Getting to the Answer: Rearrange the equations to be in the same form, with the y terms before the x terms, and then add:

The correct answer is (D).

18. C

Difficulty: Medium

Category: Substitution

Getting to the Answer: Write a system of equations where c is the cost of the couch in dollars and b is the cost of the bed in dollars. A bed costs $40 less than three times the cost of the couch, or b = 3c − 40. Together, a bed and a couch cost $700, so b + c = 700.

The system of equations is:

The top equation is already solved for b, so substitute 3c − 40 into the bottom equation for b and solve for c:

Remember to check if you solved for the right thing! The couch costs $185, so the bed costs 3($185) − $40 = $555 − $40 = $515. This means the bed costs $515 − $185 = $330 more than the couch. Therefore, (C) is correct.

19. C

Difficulty: Hard

Category: Number of Possible Solutions

Getting to the Answer: The system has infinitely many solutions, so both equations must describe the same line. Notice that if you multiply the x− and y-coefficients in the second equation by 16, you arrive at the x− and y-coefficients in the first equation. The constant k times 16 must then equal the constant in the first equation, or −32:

Therefore, (C) is correct.

20. D

Difficulty: Hard

Category: Number of Possible Solutions

Getting to the Answer: Rearrange the equations and write them on top of each other so that the x− and y-terms line up:

In a system of equations that has no solution, the x-­coefficients must equal each other and the y-coefficients must equal each other, but the constant on the right needs to be different. Thus, for the x-coefficients, 36 = 6b and b = 6. For the y-coefficients, a = −7. The question asks for the value of |a − b|, which is |−7 − 6| = |−13| = 13, choice (D).

21. 1/2 or .5

Difficulty: Medium

Category: Combination

Getting to the Answer: Start by clearing the fractions from the first equation (by multiplying by 8) to make the numbers easier to work with. Then, use combination to solve for y:

Take one-fourth of 2 to get , then grid in 1/2 or .5.

22. C

Difficulty: Medium

Category: Word Problems

Getting to the Answer: Because the variables are defined in the question stem and because the answer choices contain the variables, the only thing left for you to do is to figure out how they relate to one another. There will be two equations: one involving the total number of aircraft that landed and one involving the total amount of landing fees collected. Add together both types of aircraft to get the total number of aircraft that landed: c + p = 312. Think carefully about which type of plane should be associated with which fee to get the latter. Commercial airliners are much more expensive; hence, your second equation should be 281c + 31p = 47,848. Only (C) contains both of those equations.

23. A

Difficulty: Medium

Category: Combination

Getting to the Answer: Choose intuitive letters for the variables: s for the small bags, L for the large bags. You’re given the cost of each, as well as the number of each sold and the total revenue generated. Next, write the system of equations that represents the information given:

Multiplying the top equation by −15 allows you to solve for s using combination:

Solving for s gives 20, which eliminates (B) and (C). Plugging this value back into the first equation allows you to find L, which is 7. Choice (A) is correct.

24. D

Difficulty: Medium

Category: Substitution

Getting to the Answer: Because x has a coefficient of 1 in the second equation, solve the system using substitution. First, solve the second equation for x to get x = 6y + 10. Then, substitute the resulting expression for x into the first equation and solve for y:

Next, substitute this value back into x = 6y + 10 and simplify:

Finally, subtract x − y to find that (D) is correct:

25. C

Difficulty: Easy

Category: Word Problems

Getting to the Answer: Translate English into math. One equation should represent the total number of meals ordered, while the other equation should represent the cost of the meals.

The number of people who ordered chicken plus the number who ordered vegetarian equals the total number of people, 62, so one equation is c + v = 62. This means you can eliminate (A). Now, write the cost equation: the cost per chicken dish, $12.75, times the number of dishes, c, plus the cost per vegetarian dish, $9.50, times number of dishes, v, equals the total bill, $725.25. The cost equation should be 12.75c + 9.5v = 725.25. Together, these two equations form the system in (C).

26. C

Difficulty: Medium

Category: Combination

Getting to the Answer: Translate English into math to write a system of equations with t being the cost of a turkey burger and w equaling the cost of a bottle of water. The first statement is translated as 2t + w = $3.25 and the second as 3t + w = $4.50. Now, set up a system:

You could solve the system using substitution, but combination is quicker in this question because subtracting the first equation from the second eliminates w and you can solve for t:

Substitute this value for t in the first equation and solve for w:

Two bottles of water would cost 2 × $0.75 = $1.50, which is (C).