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Course: Digital SAT Math > Unit 10

Lesson 3: Linear relationship word problems: advanced

Understanding linear relationships | Lesson

Google Classroom

A guide to understanding linear relationships on the digital SAT

What are linear relationships?

A linear relationship is any relationship between two variables that creates a line when graphed in the

x y

-plane. Linear relationships are very common in everyday life.

In this lesson, we'll:

Review the basics of linear relationships
Practice writing linear equations based on word problems
Identify the important features of linear functions

The skills covered here will be important for the following SAT lessons:

Graphs of linear equations and functions
Systems of linear equations word problems
Linear inequality word problems
Graphs of linear systems and inequalities

You can learn anything. Let's do this!

Linear relationships

Linear equations can be used to represent the relationship between two variables, most commonly

x

and

y

. To form the simplest linear relationship, we can make our two variables equal:

y = x

By plugging numbers into the equation, we can find some relative values of

x

and

y

$x$ ‍	$y$ ‍
$0$ ‍	$0$ ‍
$1$ ‍	$1$ ‍
$2$ ‍	$2$ ‍
$3$ ‍	$3$ ‍

If we plot those points in the

x y

-plane, we create a line.

Every possible linear relationship is just a modification of this simple equation. We might multiply one of the variables by a coefficient or add a constant to one side of the equation, but we'll still be creating a linear relationship.

How do we translate word problems into linear equations?

Modeling real world scenarios

Khan Academy video wrapper

Modeling with linear equations: gym membership & lemonadeSee video transcript

Translating word problems

It may not be hard to translate "Maya is

3

inches taller than Geoff" into a linear equation, but some SAT word problems are several sentences long, and the information we need to build an equation may be scattered around.

Let's look at some examples!

A car with a price of

$ 17,000

is to be purchased with an initial payment of

$ 5,000

and monthly payments of

$ 240

. Which of the following equations can be used to find the number of monthly payments,

m

, required to complete the purchase, assuming there are no taxes or fees?

We're given three important values here:

17,000

5,000

, and

240

$$ 17,000$ ‍ is the total price, so that's what everything else needs to add up to.
$$ 5,000$ ‍ is a one-time payment.
$$ 240$ ‍ is a constant amount that's paid every month, so it needs to be multiplied by $m$ ‍, the number of months.

The total price,

$ 17,000

, equals the sum of the other payments: the initial

$ 5,000

payment and the

$ 240

paid each month (

m

17,000 = 5,000 + 240 m

The width of a rectangular vegetable garden is

w

feet. The length of the garden is

8

feet longer than its width. Which of the following expresses the perimeter, in feet, of the vegetable garden in terms of

w

We're only provided one value here, but the context provides us the rest of the information we need:

The width of the garden is $w$ ‍ feet.
The length of the garden ( $ℓ$ ‍) is $8$ ‍ feet more than the width, so $w + 8$ ‍ feet.
The perimeter of a rectangle, $P$ ‍, is the sum of its two lengths and two widths ( $w$ ‍): $P = 2 ℓ + 2 w$ ‍.

We can write the equation for the perimeter and substitute

w + 8

for

ℓ

\begin{aligned} P & = 2 ℓ + 2 w \\ = 2 (w + 8) + 2 w \\ = 2 w + 16 + 2 w \\ = 4 w + 16 \end{aligned}

The perimeter is equal to the sum of

16

and

4

times the width.

P = 4 w + 16

The concession stand at a high school baseball game sold bags of peanuts for

$ 2.50

each and hot dogs for

$ 3.00

each. If the concession stand brought in

$ 196

and sold

42

hot dogs, how many bags of peanuts did the concession stand sell?

We have a number of important values to keep track of here:

$$ 196$ ‍ is the total amount of money, so that's what everything else needs to add up to.
$$ 3.00$ ‍ is the money from each hot dog sold.
$42$ ‍ is the number of hot dogs sold, so we'll need to multiply that by $$ 3.00$ ‍ to find the amount of money from selling all of them.
$$ 2.50$ ‍ is the money from each bag of peanuts sold.
The only thing we don't know is how many bags of peanuts were sold, so that's going to be our only variable. Let's call it $p$ ‍.

We can write an equation that represents this relationship:

\begin{aligned} total money & = hot dog money + peanuts money \\ 196 & = 3.00 (42) + 2.50 (p) \end{aligned}

The total revenue of

$ 196

is equal to the sum of the revenue from hot dogs and the revenue from bags of peanuts. We can simplify and solve for

p

\begin{aligned} 196 & = 126 + 2.50 p \\ 70 & = 2.50 p \\ 28 & = p \end{aligned}

The concession stand sold $28$ ‍ bags of peanuts.

What will we be asked to do in linear equations word problems?

On the test, we may be asked to:

Write our own equation based on the word problem
Write our own equation and then solve it
Solve a given equation based on the word problem

Try it!

Try: identify parts of a linear equation

A helicopter, initially hovering

35

feet above the ground, begins to ascend at a speed of

16

feet per second. Write an equation that can be used to find

t

, the number of seconds it takes for the helicopter to reach

179

feet above the ground.

The total height, which everything else must add up to, is

feet.

The starting height of the helicopter is

feet.

The amount of time it takes is

seconds.

We can write the equation as

179 = 35 + 16 t

What are important features of linear functions?

Linear equations in slope-intercept form

Khan Academy video wrapper

Constructing linear equations from contextSee video transcript

Linear functions

Any linear equation with two variables is technically a function. Linear functions are usually written in either slope-intercept form or standard form. We need a thorough and flexible understanding of these forms in order to approach many SAT questions about linear relationships.

Slope-intercept form

The slope-intercept form of a linear function,

y = m x + b

, where

m

and

b

are constants, tells us both the slope and the

y

-intercept of the line:

The slope is equal to $m$ ‍.
Slope describes the rate of change in a linear relationship.
On a graph, slope is the direction and steepness of the line.
In a word problem, slope is the value that's applied multiple times, for instance a monthly payment on a large purchase.
Words like "rate", "per", and "each" are some clues that we're looking at a value that's the slope.
If we have any two points from a linear relationship, we can calculate slope by dividing the change in $y$ ‍ by the change in $x$ ‍:
$slope = \frac{change in y}{change in x} = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ ‍
The $y$ ‍-intercept is equal to $b$ ‍.
The $y$ ‍-intercept of a line is the $y$ ‍-value when $x = 0$ ‍.
On a graph, the $y$ ‍-intercept is the point where the line crosses the $y$ ‍-axis.
In a word problem, the $y$ ‍-intercept represents a constant value applied one time, for instance the initial down payment on a large purchase.
Words like "initial", "starting", and "one-time" are some clues that we're looking at a value that's the $y$ ‍-intercept.
If we know the slope of a linear relationship, we can plug any point into the linear equation and solve for the $y$ ‍-intercept.

Standard form

The standard form of a linear function,

A y + B x = C

, where,

A

B

, and

C

are constants, will often be used in word problem scenarios that have two inputs, instead of an input and an output. To find the slope or

y

-intercept of a line in standard form, it's often most convenient to convert the equation to slope-intercept form by isolating

y

What will we be asked to do in linear function word problems?

On the test, we may be asked to:

Write our own linear function based on the word problem (We may need to calculate the slope or $y$ ‍-intercept in more challenging questions.)
Identify the meaning of a value in a given function that models a scenario

Try it!

Try: build a linear function

**Shipping Charges**
Merchandise weight (pounds)	Shipping charge
$5$ ‍	$$ 16.49$ ‍
$10$ ‍	$$ 23.99$ ‍
$25$ ‍	$$ 46.49$ ‍

The table above shows shipping charges for an online retailer that sells used textbooks. There is a linear relationship between the shipping charge and the weight of the merchandise. Write a function in slope-intercept form that relates

y

, the shipping charge in dollars, and

x

, the merchandise weight in pounds.

The slope of the function represents the

and is

The

y

-intercept of the function represents the

and is

The function is:

Things to remember

slope = \frac{change in y}{change in x} = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}

The slope-intercept form of a linear equation,

y = m x + b

, tells us both the slope and the

y

-intercept of the line:

The slope is equal to $m$ ‍.
The $y$ ‍-intercept is equal to $b$ ‍.

We can write the equation of a line as long as we know either of the following:

The slope of the line and a point on the line
Two points on the line

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Digital SAT Math

Course: Digital SAT Math > Unit 10

What are linear relationships?

Linear relationships

How do we translate word problems into linear equations?

Modeling real world scenarios

Translating word problems

Let's look at some examples!

What will we be asked to do in linear equations word problems?

Try it!

What are important features of linear functions?

Linear equations in slope-intercept form

Linear functions

Slope-intercept form

Standard form

What will we be asked to do in linear function word problems?

Try it!

Your turn!

Things to remember

Want to join the conversation?