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### Course: Digital SAT Math > Unit 2

Lesson 3: Linear relationship word problems: foundations# Understanding linear relationships | Lesson

A guide to understanding linear relationships on the digital SAT

## What are linear relationships?

A $xy$ -plane. Linear relationships are very common in everyday life.

**linear relationship**is any relationship between two variables that creates a line when graphed in theIn 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:

By plugging numbers into the equation, we can find some relative values of $x$ and $y$ .

If we plot those points in the $xy$ -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

### 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 $\mathrm{\$}\mathrm{17,000}$ is to be purchased with an initial payment of $\mathrm{\$}\mathrm{5,000}$ and monthly payments of $\mathrm{\$}240$ . 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?

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 $w$ ?

The concession stand at a high school baseball game sold bags of peanuts for $\mathrm{\$}2.50$ each and hot dogs for $\mathrm{\$}3.00$ each. If the concession stand brought in $\mathrm{\$}196$ and sold $42$ hot dogs, how many bags of peanuts did the concession stand sell?

#### 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!

## What are important features of linear functions?

### Linear equations in slope-intercept form

### 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 ${y=mx+b}$ , where $m$ and $b$ are constants, tells us both the slope and the $y$ -intercept of the line:

**slope-intercept form**of a linear function,- The slope is equal to
.$m$ - The
-intercept is equal to$y$ .$b$

#### Standard form

The ${Ay+Bx=C}$ , 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 $y$ .

**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!

## Your turn!

## Things to remember

The $y=mx+b$ , tells us both the slope and the $y$ -intercept of the line:

**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

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thank you !(65 votes)- how many months and hours you used to practice and study foe each section?(12 votes)

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