How much do you know? - Linear equations and graphs - The heart of algebra

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

How much do you know?
Linear equations and graphs
The heart of algebra

LEARNING OBJECTIVES

After completing this chapter, you will be able to:

· Isolate a variable

· Translate word problems into equations

· Calculate the slope of a line given two points

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

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

· Describe the slopes of parallel and perpendicular lines

110/600 SmartPoints® (Very High Yield)

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.

image

1. Which value of x satisfies the equation above?

A. −61

B.−55

C.−41

D. −35

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

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

B.c = 1.06(54.95d) + 10

C.c = 1.06(54.95d + 10)

D. c = 1.06(54.95 + 10)d

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

A. xy + 6

B.yx + 6

C.x + y + 6

D. y + x − 6

Price of One Can

Projected Number of Cans Sold

$0.75

10,000

$0.80

9,000

$0.85

8,000

$0.90

7,000

$0.95

6,000

$1.00

5,000

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

A. n = −20,000p + 25,000

B.n = −200p + 250

C.n = 200p + 250

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

image

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

A. $37.00

B.$48.00

C.$62.50

D. $74.50

Check Your Work

1. A

Difficulty: Easy

Category: Solving Equations

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

2. B

Difficulty: Medium

Category: Word Problems

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

3. D

Difficulty: Medium

Category: Word Problems

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

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

image

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

You could also pick numbers to answer this question.

4. A

Difficulty: Medium

Category: Linear Graphs

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

image

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

5. D

Difficulty: Hard

Category: Linear Graphs

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

Solving equations

Learning objective

After this lesson, you will be able to:

· Isolate a variable

To answer a question like this:

image

Which of the following values is equal to x?

A. −36

B. 4

C. 12

D. 20

You need to know this:

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

You need to do this:

It usually makes sense to proceed in this order:

1. Eliminate any fractions.

2. Collect and combine like terms.

3. Divide to leave the desired variable by itself.

Explanation:

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

image

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

image

Next, subtract 5x from both sides:

image

Finally, subtract 24 from both sides:

image

Therefore, (A) is correct.

Try on Your Own

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

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

image

1. image Which approximate value of x satisfies the equation shown?

A. 4.3

B.4.6

C.6.6

D. 6.8

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

2. If image, then what does image equal?

image

image

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

A. image

B.image

C.image

D. image

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

A. −3

B.3

C.6

D. 12

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

5. In the equation image, which of the following is a possible value of x − 4?

A. image

B.4

C.image

D. 5