Systems of equations - Heart of algebra - Math

PSAT/NMSQT Prep 2019 - Princeton Review 2019

Systems of equations
Heart of algebra
Math

CHAPTER OBJECTIVES

By the end of this chapter, you will be able to:

1. Distinguish between independent and dependent equations

2. Solve two-variable systems of linear equations

3. Determine the most efficient way to solve systems of equations

4. Translate word problems into multiple equations

SMARTPOINTS

Point Value

SmartPoint Category

50 Points

Systems of Linear Equations

Prepare

SYSTEMS OF EQUATIONS

The linear equations detailed in the previous chapter are well suited for modeling a variety of scenarios and for solving for a single variable in terms of another that is clearly defined (e.g., What is the cost of a data plan if you consume 4 GB of data in a month?). However, sometimes you will be given a set of multiple equations with multiple variables that are interdependent. For example, suppose a $50/month cell phone plan includes $0.05 text messages and $0.40 voice calls, with a cap of 1,000 combined text messages and voice calls.

This scenario can be represented by the following system of equations:

Solving such a system would enable you to determine the maximum number of text messages and voice calls you could make under this plan, while optimizing total usage. To solve systems of equations, you’ll need to rely on a different set of tools that builds on the algebra you’re already familiar with. The following question shows an example of such a system in the context of a test-like question:

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

1. 0

2. 4

3. 6

4. 12

You might be tempted to switch on math autopilot at this point and employ substitution, solving the second equation for s in terms of r:

s = 12 — r

You could plug the resulting expression back into the other equation and eventually solve for r, but remember, the PSAT tests your ability to answer math questions in the most efficient way. The following table contains some strategic thinking designed to help you find the most efficient way to answer this question on Test Day, along with some suggested scratchwork.

Strategic Thinking

Math Scratchwork

Step 1: Read the question, identifying and organizing important information as you go

In this case, you’re looking for the value of r. There are two equations that involve r and s.

3r + 2s = 24

r + s = 12

Step 2: Choose the best strategy to answer the question

Is there any way you can make the first equation look like the second one? Does the quantity r + s exist in the first equation in some form?

How can you effectively use both equations?

Once you’ve written the first equation in terms of r + s, substitute the value of r + s (which is 12) into the second equation and solve for r.

3r + 2s = 24

r + 2r + 2s = 24

r + 2(r + s) = 24

r + 2(12) = 24

r = 0

Step 3: Check that you answered the right question

Be careful! The question isn’t asking for the value of r. Add 6 to your result and you should see that (C) is the correct answer.

r + 6 = 0 + 6

r + 6 = 6

Note

Explanations for each simplifying step are not always included in this chapter. If you get stuck, review the information on simplifying and solving equations in chapter 1.

INDEPENDENT VERSUS DEPENDENT EQUATIONS

Generally, when you have a system involving n variables, you need n independent equations to solve for those variables. Thus, if you have a system of two variables, you need two independent equations. Three variables would require three independent equations, and so on.

Systems of equations are extremely useful in modeling and simulation. Complex mathematical problems, such as weather forecasting or crowd control predictions, often require 10 or more equations to be simultaneously solved for multiple variables. Fortunately, you won’t encounter anything this daunting on Test Day.

Before we outline the process for solving two-variable systems of equations, let’s clarify one of the key requirements. Earlier, it was stated that you need two independent equations to solve for two variables, but what exactly is an independent equation? Consider the equation 4x + 2y = 8. You could use properties of equality to transform this equation in a number of different ways. For example, you could multiply both sides by 2, resulting in the equation 8x + 4y = 16.

While it seems as though we’ve just created an additional equation, this is misleading, as the second equation has the same core variables and relationships as the first equation. This is termed a dependent equation, and two dependent equations cannot be used to solve for two variables. Look what happens when we try to use substitution. Start by isolating y in the original equation; the result is y = 4 — 2x.

Substituting that into the second equation, notice what happens:

Although 16 does in fact equal 16, this doesn’t bring us any closer to solving for either of the variables. In fact, if you arrive at a result like this when solving a system of equations, then the two equations are dependent. In this case, the system has infinitely many solutions because you could choose any number of possible values for x and y.

Note

When two equations are dependent, one equation can be obtained by algebraically manipulating the other equation. Graphically, dependent equations both describe the same line in the coordinate plane and therefore have the same slope and the same y-intercept.

At other times, you’ll encounter equations that are fundamentally incompatible with each other. For example, if you have the two equations 4x + 2y = 8 and 4x + 2y = 9, it should be obvious that there are no values for x and y that will satisfy both equations at the same time. Doing so would violate fundamental laws of math. In this case, you would have a system of equations that has no solution. These two equations define parallel lines, which by definition never intersect.

Knowing how many solutions a system of equations has will tell you how graphing them in the same coordinate plane should look. Remember, the solution of a system of equations consists of the point or points where their graphs intersect.

If your system has …

then it will graph as…

Reasoning

no solution

two parallel lines

Parallel lines never intersect.

one solution

two lines intersecting at a single point

Two straight lines have only one intersection.

infinitely many solutions

a single line (one line directly on top of the other)

One equation is a manipulation of the other—their graphs are the same line.

Because you could encounter any of these three situations on Test Day, make sure you are familiar with all of them.

Let’s examine a sample question to investigate the requirements for solving a system of equations:

2.

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

Work through the Kaplan Method for Math step-by-step to solve this question. The following table shows Kaplan’s strategic thinking on the left, along with suggested math scratchwork on the right.

Strategic Thinking

Math Scratchwork

Step 1: Read the question, identifying and organizing important information as you go

You are looking for the value of z, given that the equation has infinitely many solutions.

This means that the second equation should be the same as the first equation after some kind of algebraic manipulation.


Step 2: Choose the best strategy to answer the question

Look for ways to make the first equation resemble the second.

The constant on the right gives you a strong clue: multiplying 40 by 40 gives 1,600. This also works well with the s term because multiplying by 40 yields 8s.

5q + 8s = 1,600

Step 3: Check that you answered the right question

Be careful! Choice D is a trap. The question isn’t looking for the number you’d multiply the first equation by. Instead, it’s looking for z, the coefficient of q.

Because the number in front of q in your transformed equation is 5, you know that z must also be equal to 5, making (B) the correct answer.

5q = zq

z = 5

SOLVING SYSTEMS OF EQUATIONS: COMBINATION & SUBSTITUTION

Now that you understand the requirements that must be satisfied to solve a system of equations, let’s look at some methods for solving these systems effectively. The two main methods for solving a system of linear equations are substitution and combination (sometimes referred to as elimination by addition).

Substitution is the most straightforward method for solving systems, and it can be applied in every situation. Unfortunately, it is often the longest and most time-consuming route for solving systems of equations as well. To use substitution, solve the simpler of the two equations for one variable, and then substitute the result into the other equation. You could use substitution to answer the following question, but you’ll see that there’s a quicker way: combination.

Combination involves adding the two equations together to eliminate a variable. Often, one or both of the equations must be multiplied by a constant before they are added together. Combination is often the best technique to use to solve a system of equations as it is usually faster than substitution.

Unfortunately, even though most students prefer substitution, systems of equations on the PSAT are often designed to be quickly solved with combination. To really boost your score on Test Day, practice combination as much as you can on Practice Tests and in homework questions so that it becomes second nature. As you work through the following question, be sure to think about which strategy will be more efficient:

3. 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

Work through the Kaplan Method for Math step-by-step to solve this question. The following table shows Kaplan’s strategic thinking on the left, along with suggested math scratchwork on the right.

Strategic Thinking

Math Scratchwork

Step 1: Read the question, identifying and organizing important information as you go

You are given a system of two equations with two unknowns and asked to find the values of a and b.





6a + 6b = 30

3a + 2b = 14

Step 2: Choose the best strategy to answer the question

Remember, while substitution could be used to answer this type of question, combination will often be faster.

What transformation will enable you to add the equations and eliminate a variable?

Combination often requires you to multiply one of your equations by a constant. In this case, notice what happens if you multiply the second equation by —3.

What’s the next step in combination?

By arranging the equations vertically, you can simply add them, combining like terms along the way.

Notice that 6b + (—6b) = 0b = 0, and you’ve eliminated b from your equation. Your goal when using combination is to set the coefficient of the variable you are trying to eliminate to a number that is equal in magnitude and opposite in sign to the coefficient in the other equation. Now you can easily solve for a.

(−3)(3a +2b) = (14)(−3)

−9a − 6b = −42

−3a + 0b = −12

 −3a = −12

  a = 4

Step 3: Check that you answered the right question

Even though the question asks you for the values of a and b, each answer choice has a different value of a. There’s no need to plug back in and find the correct value of b. Choice (B) is correct.


Combination can also be used when the test makers ask you for the sum or difference of two variables, as in the following question:

4. If 5c − 2b = 15 and 3b − 4c = 12, what is the value of b + c ?

1. −27

2. −3

3. 8

4. 27

Work through the Kaplan Method for Math step-by-step to solve this question. The following table shows Kaplan’s strategic thinking on the left, along with suggested math scratchwork on the right.

Strategic Thinking

Math Scratchwork

Step 1: Read the question, identifying and organizing important information as you go

You are being asked to find the value of b + c. The question stem provides two equations involving b and c.

5c — 2b = 15

3b — 4c = 12

Step 2: Choose the best strategy to answer the question

How can you quickly and accurately answer the question? Why are the test makers asking for the quantity b + c and not the values of b and c independently?

The fact that you’re solving for b + c suggests that there’s a time-saving shortcut to be found. Because you’re not trying to get rid of a variable, see if you can add the equations to get a result that has b + c equal to some numerical value. Before you add, don’t forget to write the variable terms in the same order for each equation.

Step 3: Check that you answered the right question

Because you’re asked to find the value of b + c, there’s nothing more to do here. Choice (D) is correct.

b + c = 27

That was much easier and faster than substitution. With substitution, you could spend more than two minutes solving a question like this. However, a bit of analysis and combination gets the job done in much less time.

TRANSLATING WORD PROBLEMS INTO MULTIPLE EQUATIONS

While solving systems of equations can be relatively straightforward once you get the hang of it, sometimes you’ll encounter a complex word problem and will need to translate it into a system of equations and then solve. It sounds a lot scarier than it actually is. Remember to use the Kaplan Strategy for Translating English into Math to set up your equations, and then solve using either substitution or combination.

Note

The Kaplan Strategy for Translating English into Math can be found in the previous chapter.

Let’s take a look at an example:

5. At a snack stand, hot dogs cost $3.50 and hamburgers cost $5. If the snack stand sold 27 snacks and made $118.50 in revenue, how many hot dogs were sold? How many hamburgers?

1. 16 hot dogs; 11 hamburgers

2. 16 hot dogs; 16 hamburgers

3. 11 hot dogs; 14 hamburgers

4. 11 hot dogs; 16 hamburgers

Work through the Kaplan Method for Math step-by-step to solve this question. The following table shows Kaplan’s strategic thinking on the left, along with suggested math scratchwork on the right.

Strategic Thinking

Math Scratchwork

Step 1: Read the question, identifying and organizing important information as you go

You need to find the number of hot dogs and hamburgers sold.


Step 2: Choose the best strategy to answer the question

Use the Kaplan Strategy for Translating English into Math.

What variables should you use?

Because both snacks start with h, that’s likely to be a confusing choice. Instead, use d for hot dogs and b for hamburgers.

How do you break apart the question into smaller phrases?

Break off each piece of relevant information into a separate phrase.

What should you do with the phrases?

Translating each phrase into a math expression will create the components of a system of equations.

What is the system of equations that will get you to the answer?

Assemble the math expressions you have to get the system of equations needed.

How can you best solve this system of equations?

In this case, you can use either substitution or combination to arrive at the numbers of hot dogs and hamburgers. To use combination, multiply the first equation by —5 to set it up.

Continuing to solve, you see that d = 11.

What does this information enable you to do?

That immediately eliminates A and B. You now can plug 11 back into the first equation to get b.

d = hot dogs sold

b = hamburgers sold

hot dogs cost $3.50 → 3.5d

hamburgers cost $5 → 5b

snack stand sold 27 snacks

d + b = 27

made $118.50 in revenue

→ Total $ = 118.5











d
+ b = 27

3.5d + 5b = 118.5

d + b = 27

11 + b = 27

b = 16

Step 3: Check that you answered the right question

The only answer choice that meets these criteria is (D).


Watch out for A, a trap answer designed to catch students who switched the variables, possibly due to choosing an ambiguous letter such as h. Choosing descriptive variable names might sound silly, but in the high-stakes environment of the PSAT, doing this can make the difference between a decent score and a National Merit Scholarship.

Note

Always choose variable names that make sense to you. Countless students struggle on multi-part questions due to disorganized notes. Don’t let that happen to you. Move beyond x and y when selecting variable names.

Other questions of this type will simply ask you to choose from a series of answer choices that describes the system of equations—they won’t actually ask you to calculate a solution! These questions can be great time-savers. Consider the following example:

6. A local airport has separate fees for commercial airliners and private planes to take into account the different rates of wear that each has on the facility. 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. c + p = 47,848; 281c + 31p = 312

2. c + p = 312; 31c + 281p = 47,848

3. c + p = 312; 281c + 31p = 47,848

4. c + p = 47,848; 31c + 281p = 312

Work through the Kaplan Method for Math step-by-step to solve this question. The following table shows Kaplan’s strategic thinking on the left, along with suggested math scratchwork on the right.

Strategic Thinking

Math Scratchwork

Step 1: Read the question, identifying and organizing important information as you go

You’re asked for the system of equations that best represents the situation given.


Step 2: Choose the best strategy to answer the question

What system of equations best models this scenario? What two equations should you construct?

You are given information about the total landing fees and the total number of planes. What quantities would add up to give you these totals?

What quantities combine to give you the total number of planes that landed at the airport?

The number of private planes and airliners combined would yield the total number of planes. This immediately eliminates A and D.

The cost for all of the commercial airliners could be represented by the cost of each airliner times the number of airliners.

Similarly, the cost for all the private planes could be represented by the cost of each plane times the number of planes.

The total amount of landing fees would then be both of these quantities added together.

Which answer choice has both of these equations in it?



A total of 312 planes landed at the airport.

$47,848 in landing fees were collected.









p
+ c = 312



commercial = 281c







private = 31p



47,848 = 281c + 31p

Step 3: Check that you answered the right question

If you chose (C), you’re absolutely correct.


Be careful! Choice B is close but switches the fee structure, drastically overcharging the private planes. Always pay close attention to the differences between answer choices to avoid traps on Test Day.

Practice

Now you’ll have a chance to try a few more test-like questions. Use the scaffolding as needed to guide you through the question and get the correct answer.

Some guidance is provided, but you’ll need to fill in the missing parts of explanations or the step-by-step math to get to the correct answer. Don’t worry—after going through the examples at the beginning of this chapter, these questions should be completely doable. If you find yourself struggling, however, review the worked examples again.

7. 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

The following table can help you structure your thinking as you go about answering this question. The Kaplan strategic thinking is provided, as are bits of structured scratchwork. If you’re not sure how to approach a question like this, start at the top and work your way down.

Strategic Thinking

Math Scratchwork

Step 1: Read the question, identifying and organizing important information as you go

You’re asked how many more texts than pictures were sent.


Step 2: Choose the best strategy to answer the question

Use the Kaplan Strategy for Translating English into Math.

Pick variables that match the context by using the letter that each word starts with.

Next, break off each piece of relevant information into a separate phrase.

What should you do with the phrases?

Translating each phrase into a math expression will get you the components of a system of equations.

How can you best solve this system of equations?

You can use either substitution or combination to arrive at the text and picture counts. Remember to think critically about which approach would be faster in this situation.

How many texts were sent? How many pictures?

Solve for each variable.

Are you done?

You’re not quite done. The question asks how many more texts were sent than pictures, so find the difference.

texts: ___ pictures: ___


text cost: _____

picture cost: _____

total texts + pictures: _____

total bill: _____



texts: ____

pictures: ____


Step 3: Check that you answered the right question

If you answered (A), you’re absolutely correct.


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

image

Larger numbers don’t make this question any different; just be careful with the arithmetic. Again, the following table can help you structure your thinking as you go about answering this question. Kaplan’s strategic thinking is provided, as are bits of structured scratchwork. If you’re not sure how to approach a question like this, start at the top and work your way down.

Strategic Thinking

Math Scratchwork

Step 1: Read the question, identifying and organizing important information as you go

You’re asked to find the product of x and y.


Step 2: Choose the best strategy to answer the question

What’s the quickest route to the answer?

You have coefficients on all four variable terms and large constants on the right sides of the equations, so combination will likely be faster than substitution.

How do you cancel out one of the variables?

Although you can often get away with manipulating only one equation, you’ll need to adjust both here. The x coefficients are 2 and 5. No integer will multiply by 2 to get 5 and vice versa, but what about multiplying both equations by numbers that will get you a common multiple on both x terms? Don’t forget to make one of them negative.

What is the next step?

Carry out combination as usual, being especially careful with the larger numbers.

What’s the value of y?

Straightforward algebra from your combined equations should reveal the value of y.

How about x?

Plug your y value back into one of the original equations and solve for x.

Lastly, multiply x and y together.

2x + 5y = 49

5x + 3y = 94

___ ( ____ + ____ = ____ )

___ ( ____ + ____ = ____ )

y = ____

x = ____

xy = ____ × ____ = ____

Step 3: Check that you answered the right question

If your answer is 51, you’re correct!



xy
= ____

Perform

Now that you’ve seen the variety of ways in which the PSAT can test you on systems of linear equations, try the following questions to check your understanding. Give yourself 4.5 minutes to tackle the following three questions:

9.

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

1.

2.

3. 8

4. 10

10.If x and y are both integers such that x + 6 = 17 and y + 9 = 12, what is the value of x + y ?

image

11.Sixty people attended a concert. Children’s tickets sold for $8 each, and adults’ tickets sold for $12 each. If $624 was collected in ticket money, what is the product of the number of children and the number of adults who attended the concert?

1. 275

2. 779

3. 864

4. 900

On your own

1. Guests at a wedding had two meal choices, chicken or 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.

2.

What is the y-coordinate of the solution to the system of equations shown?

1. —1

2. 0

3.

4. 5

3.

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

1.

2.

3.

4.

4. 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

5. 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

6.

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

7.

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

1.

2.

3. 2

4. 12

8. 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. 12

2. 15

3. 18

4. 21

9. image

If (x, y) is the solution to the system of equations graphed in the figure, what is the value of x + y ?

1. —2

2. —1

3. 1

4. 4