## What are linear relationships?

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

**Maya is **

We can express the *linear relationship* between Maya's height in inches,

**Tai runs **

We can express the *linear relationship* between the distance Tai runs in miles,

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

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

If we plot those points in the

**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 & lemonade

See video transcript

### Translating word problems

It may not be hard to translate "Maya is

#### Let's look at some examples!

A car with a price of

We're given three important values here:

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

The total price,

The width of a rectangular vegetable garden is

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

We can write the equation for the perimeter and substitute

The perimeter is equal to the sum of

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

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

is the$\mathrm{\$}196$ *total*amount of money, so that's what everything else needs to add up to. is the money from each hot dog sold.$\mathrm{\$}3.00$ is the number of hot dogs sold, so we'll need to multiply that by$42$ to find the amount of money from selling all of them.$\mathrm{\$}3.00$ is the money from each bag of peanuts sold.$\mathrm{\$}2.50$ - 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:

The total revenue of

**The concession stand sold **

#### 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

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

## What are important features of linear functions?

### Linear equations in slope-intercept form

Khan Academy video wrapper

Constructing linear equations from context

See 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,

- 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

by the change in$y$ :$x$ $\text{slope}={\displaystyle \frac{\text{change in}y}{\text{change in}x}}={\displaystyle \frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}}$ - The
-intercept is equal to$y$ .$b$ The

of a line is the -intercept$y$ -value when$y$ .$x=0$ - On a graph, the
-intercept is the point where the line crosses the$y$ -axis.$y$ - In a word problem, the
-intercept represents a constant value applied one time, for instance the initial down payment on a large purchase.$y$ - Words like "initial", "starting", and "one-time" are some clues that we're looking at a value that's the
-intercept.$y$

If we know the slope of a linear relationship, we can plug any point into the linear equation and solve for the

-intercept.$y$ - On a graph, the

#### Standard form

The **standard form** of a linear function,

#### 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
-intercept in more challenging questions.)$y$ - Identify the meaning of a value in a given function that models a scenario

### Try it!

Try: build a linear function

Merchandise weight (pounds) | Shipping charge |
---|---|

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

The slope of the function represents the

and is

.

The

and is

.

The function is:

## Your turn!

Practice: write a linear equation

Tamika purchases a new mattress for

Practice: solve a linear equation

Lawrence will mix

Practice: interpret a Linear function

The equation above models

Practice: Linear function word problems

A farm purchased a combine harvester valued at

## Things to remember

The **slope-intercept form** of a linear equation,

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

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