Systems of linear equations - The heart of algebra

PSAT/NMSQT Prep 2020 - Princeton Review 2020

Systems of linear equations
The heart of algebra

Learning Objectives

After completing this chapter, you will be able to:

· Solve systems of linear equations by substitution

· Solve systems of linear equations by combination

· Determine the number of possible solutions for a system of linear equations, if any

45/600 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.

1. What is the value of x for the given equations above?

1. −3

2. 0

3. 3

4. 5

2. A television set costs $25 less than twice the cost of a radio. If the television and radio together cost $200, how much more does the television cost than the radio?

1. $50

2. $75

3. $100

4. $125

3. At a snack stand, hot dogs cost $3.50 and hamburgers cost $5.00. If the snack stand sold 27 snacks and made $118.50 in revenue, how many hot dogs and how many hamburgers were sold?

1. 16 hot dogs; 11 hamburgers

2. 16 hot dogs; 16 hamburgers

3. 11 hot dogs; 14 hamburgers

4. 11 hot dogs; 16 hamburgers

4. A certain student cell phone plan charges $0.10 per text and $0.15 per picture, with no additional monthly fee. If a student sends a total of 75 texts and pictures in one month and is billed $8.90 for that month, how many more texts did he send than pictures?

1. 19

2. 28

3. 36

4. 47

5. In the system of linear equations shown, z represents a constant. If the system of equations has infinitely many solutions, what is the value of z ?

1.

2. 5

3. 8

4. 40

6.

Check Your Work

1. A

Difficulty: Easy

Category: Substitution

Getting to the Answer: Solve the second equation for y in terms of x (which yields y = —x), then substitute into the first equation and solve:

Choice (A) is correct.

2. A

Difficulty: Medium

Category: Substitution

Getting to the Answer: Translate English into math to write a system of equations with r as the cost of the radio in dollars and t as the cost of the television in dollars. A television costs $25 less than twice the cost of the radio, or t = 2r − 25. Together, a radio and a television cost $200, so r + t = 200.

The system of equations is:

The top equation is already solved for t, so substitute 2r − 25 into the second equation for t:

The radio costs $75, so the television costs 2(75) − 25 = 150 − 25 = $125. This means the television costs $125 − $75 = $50 more than the radio, which is (A).

3. D

Difficulty: Medium

Category: Combination

Getting to the Answer: Begin by translating English into math. Define the variables logically: d for hot dogs, b for hamburgers. You’re given the cost of each, as well as the number of snacks sold and the total revenue generated. Next, write the system of equations that represents the information given:

Multiplying the top equation by −5 allows you to solve for d using combination:

Dividing both sides by −1.5 gives d = 11, which eliminates (A) and (B). Plugging 11 in for d in the first equation in the system gives you 11 + b = 27. Subtract 11 from both sides to find that b = 16. (D) is correct.

4. A

Difficulty: Medium

Category: Combination

Getting to the Answer: Translate English into math to make sense of the situation. First, define your variables: t for texts and p for pictures are good choices. You know that this student sent a total of 75 texts and pictures. You’re also told each text costs $0.10 and each picture is $0.15, and that the bill is $8.90. You’ll have two equations: one relating the numbers of texts and pictures, and a second relating the costs associated with each:

Multiplying the second equation by 10 allows you to solve for p using combination:

Subtract the second equation from the first to find that −0.5p = −14 and p = 28. But you’re not done yet; you’re asked for the difference between the text and picture count. Substitute 28 for p in the first equation and then solve for t to get t = 47. Subtracting 28 from 47 yields 19, which is (A).

5. B

Difficulty: Medium

Category: Number of Possible Solutions

Getting to the Answer: A system of equations that has infinitely many solutions results when you can algebraically manipulate one equation to arrive at the other. Examining the right sides of the equations, you see that 40 × 40 = 1,600; therefore, multiplying the first equation by 40 will give 1,600 on the right: 5q + 8s = 1,600. The first equation is now identical to the second equation, meaning z must be 5, which is (B).

Substitution

Learning Objective

After this lesson, you will be able to:

· Solve systems of linear equations by substitution

To answer a question like this:

1. What is the value of y if 5x + 3y = 20 and x + y = 20 ?

1. −40

2. −20

3. 20

4. 40

You need to know this:

A system of two linear equations simply refers to the equations of two lines. “Solving” a system of two linear equations usually means finding the point where the two lines intersect. (However, see the lesson titled “Number of Possible Solutions” later in this chapter for exceptions.)

There are two ways to solve a system of linear equations: substitution and combination. For some PSAT questions, substitution is faster; for others, combination is faster. We’ll cover combination in the next lesson.

You need to do this:

To solve a system of two linear equations by substitution, do the following:

· Isolate a variable (ideally, one whose coefficient is 1) in one of the equations.

· Substitute the result into the other equation.

Explanation:

Isolate x in the second equation, then substitute the result into the first equation:

Thus, (D) is correct. If you needed to know the value of x as well, you could now substitute 40 for y into either equation to find that x = −20.

Try on Your Own

Directions

Solve these questions using substitution. 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. Clarice had twice as many nickels as dimes in her piggy bank. When she adds 4 more nickels, she has three times as many nickels as dimes. What was the total number of nickels and dimes in Clarice’s piggy bank before she added the additional nickels?

1. 4

2. 8

3. 12

4. 16

2. HINT: Ask yourself: Which variable in Q2 is the easier one to isolate?

3. What is the value of b that satisfies 5c + 5b = 20 and 5bc = 4 ?

4. What is the value of xy from the solution of the above system of equations?

1. —5

2. 0

3. 2

4. 5

5. HINT: Since the correct answer to Q4 requires you to know the value for r, solve for and substitute s in terms of r.

6. If 3r + 2s = 24 and r + s = 12, what is the value of r + 6 ?

1. 0

2. 4

3. 6

4. 12

7. At a certain restaurant, there are 25 tables and each table has either 2 or 4 chairs. If a total of 86 chairs accompany the 25 tables, how many tables have exactly 4 chairs?

1. 7

2. 12

3. 15

4. 18

8.

Combination

Learning Objective

After this lesson, you will be able to:

· Solve systems of linear equations by combination

To answer a question like this:

1.

If the lines represented by the equations above intersect at the point (x, y), then what is the value of y ?

1. —3

2. —2

3. 2

4. 3

You need to know this:

Combining two equations means adding or subtracting them, usually with the goal of either eliminating one of the variables or solving for a combination of variables (e.g., 5n + 5m).

You need to do this:

To solve a system of two linear equations by combination, do the following:

· Make sure that the coefficients for one variable have the same absolute value. (If they don’t, multiply one equation by an appropriate constant. Sometimes, you’ll have to multiply both equations by constants.)

· Either add or subtract the equations to eliminate one variable.

· Solve for the remaining variable, then substitute its value into either equation to solve for the remaining variable.

Explanation:

Both variables have different coefficients in the two equations, but you can convert the 3x in the second equation to 6x by multiplying the entire second equation by 2:

Now that the coefficients for one variable are the same, subtract the second equation from the first to eliminate the x variable. (Note that if the x-coefficients were 6 and −6, you would add the equations instead of subtracting.)

Solve this equation for y:

(A) is the correct answer. If the question asked for x instead of y, you would now substitute −3 into either of the original equations and solve for x. (For the record, x = 1.)

Try on Your Own

Directions

Solve these questions using combination. 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. What is the y-coordinate of the solution to the system of equations shown?

1. —1

2. 0

3.

4. 5

2. HINT: There’s no need to solve for b and c separately in Q7.

3. If —8c − 3b = 11 and 6b + 6c = 4, what is the value of 3b — 2c ?

1. −27

2. −3

3. 8

4. 15

4. If 6a + 6b = 30 and 3a + 2b = 14, then what are the values of a and b ?

1. a = 2; b = 2

2. a = 4; b = 1

3. a = 1; b = 4

4. a = 3; b = 1

5. Given 2x + 5y = 49 and 5x + 3y = 94, what is the product of x and y ?

image

6. Sixty people attended a concert. Children’s tickets sold for $8 each and adult tickets sold for $12 each. If $624 was collected in ticket money, how many more adults than children attended the concert?

1. 0

2. 12

3. 24

4. 60

7.

Number of Possible Solutions

Learning Objective

After this lesson, you will be able to:

· Determine the number of possible solutions for a system of linear equations, if any

To answer a question like this:

1.

In the system of linear equations above, k represents a constant. What is the value of 3k if the system of linear equations has no solution?

1. 20

2. 30

3. 60

4. 80

You need to know this:

The solution to a system of linear equations consists of the values of the variables that make both equations true.

A system of linear equations may have one solution, infinitely many solutions, or no solution.

If a system of equations represents two lines that intersect, then the system will have exactly one solution (in which the x- and y-values correspond to the point of intersection).

If a system of equations has infinitely many solutions, the two equations actually represent the same line. For example, 2x + y = 15 and 4x + 2y = 30 represent the same line. If you divide the second equation by 2, you arrive at the first equation. Every point along this line is a solution.

If a system of equations has no solution, as in the question above, the lines are parallel: there is no point of intersection.

No Solution shows two parallel lines. One solution shows two lines that intersect at one location. Infinitely Many Solutions shows one line directly on top of the other line (same line).

You need to do this:

· If the question states that the system has no solution, set both x-coefficients equal to each other and both y-coefficients equal to each other to make the lines parallel, but be sure that the y-intercepts (or constant terms, if the equations are in ax + by + c form) are different.

· If the question states that the system has infinitely many solutions, make the x-coefficients equal, the y-coefficients equal, and the y-intercepts (or constant terms) equal.

· If the question states that the system has one solution and provides the point of intersection, substitute the values at that point of intersection for x and y in the equations.

Explanation:

Start by recognizing that for two lines to be parallel, both the x-coefficients must be equal and the y-coefficients must be equal. Manipulate the second equation so that it is in the same format as the first one:

The y-coefficient in the first equation, 10x − 4y = 8, is 4. Divide the second equation by 2 in order to make the y-coefficients in both equations equal:

Now, set the x-coefficient equal to that in the first equation:

Note that the question asks for the value of 3k, so the correct answer is (C), 60.

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: How can the x- and y-values you are given as the solution to the system in Q11 help you find h and k?

2. What is the value of if the (x, y) solution of the above system of equations is (—5, 2) ?

1.

2. 2

3.

4. 6

3. HINT: For Q12, if a system of equations has infinitely many solutions, what do you know about the two equations?

4. If q is a constant and the above system of equations has infinitely many solutions, what is the value of q ?

1. —9

2.

3.

4. 9

5. HINT: For Q13, what does it mean, graphically, when a system has no solution?

6. In the system of linear equations shown above, z is a constant. If the system has no solution, what is the value of z ?

1.

2.

3. 8

4. 10

7. For which of the following values of w will the system of equations above have no solution?

1. —8

2. —4

3. 4

4. 8

8. If the system of linear equations shown above has infinitely many solutions, and c is a constant, what is the value of c ?

1.

2.

3. 2

4. 12

9.

On Test Day

Many PSAT Math questions can be solved in more than one way. A little efficiency goes a long way in helping you get through the Math sections on time, so it’s useful to try solving problems more than one way to learn which way is fastest.

Try this question using two approaches: substitution and combination. Time yourself on each attempt. Which approach allowed you to get to the answer faster?

1.

1. What is the value of x if 25x − 7y = 28 and 10x + 7y + 18 = 60 ?

1.

2.

3. 2

4.

The answer and both ways of solving 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. What is the value of yx, if 6x − 4y = 8 and 5y − 7x = 12 ?

1. —8

2. 4

3. 12

4. 20

2. A bed costs $40 less than three times the cost of a couch. If the bed and couch together cost $700, how much more does the bed cost than the couch?

1. $185

2. $225

3. $330

4. $515

3. If k is a constant and the above system of linear equations has infinitely many solutions, what is the value of k ?

1. —8

2. −4

3. −2

4. −1

4. What is the value of |ab| if a and b are constants and the above system of equations has no solution?

1. —13

2. —8

3. 8

4. 13

5. If and −4yx = 12, what is of y ?

6. A local airport has separate fees for commercial airliners and private planes. Commercial flights are charged a landing fee of $281 per flight and private planes are charged a landing fee of $31 per flight. On a given day, a total of 312 planes landed at the airport and $47,848 in landing fees was collected. Solving which of the following systems of equations yields the number of commercial airliners, c, and the number of private planes, p, that landed at the airport on the day in question?

1.

2.

3.

4.

7. At a certain coffee store, the small bag of beans costs $2.50 and the large bag of beans costs $15. If the store sold 27 small and large bags of beans and had $155 in revenue in one week, how many small bags and large bags of beans were sold?

1. 20 small bags, 7 large bags

2. 7 small bags, 20 large bags

3. 8 small bags, 19 large bags

4. 20 small bags, 9 large bags

8. If (x, y) is a solution to the system of equations shown above, then what is the value of xy ?

1.

2.

3.

4.

9. Guests at a wedding had two meal choices, chicken and vegetarian. The catering company charges $12.75 for each chicken dish and $9.50 for each vegetarian dish. If 62 people attended the wedding and the catering bill was $725.25, which of the following systems of equations could be used to find the number of people who ordered chicken, c, and the number of people who ordered vegetarian, v, assuming everyone ordered a meal?

1.

2.

3.

4.

10.Two turkey burgers and a bottle of water cost $3.25. If three turkey burgers and a bottle of water cost $4.50, what is the cost of two bottles of water?

1. $0.75

2. $1.25

3. $1.50

4. $3.00

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.

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 6a 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 5a + 8b, 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 xy 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.

1. 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.

2. 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).

3. 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.

4. 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.

5. 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).

1. 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.

2. 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.

3. 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 + 40zy = 20. Now, equate the coefficients of y to get 4y = 40zy. Solve for z to reveal that z = , 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.

4. 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.

5. 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.

1. 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.

1. 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).

2. 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.

3. 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.

4. 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 |ab|, which is |−7 − 6| = |−13| = 13, choice (D).

5. 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.

6. 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: p + c = 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.

7. 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.

8. 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 xy to find that (D) is correct:

9. 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).

10.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).