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## Digital SAT Math

### 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

**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:

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

### 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 dollar sign, 17, comma, 000 is to be purchased with an initial payment of dollar sign, 5, comma, 000 and monthly payments of dollar sign, 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 dollar sign, 2, point, 50 each and hot dogs for dollar sign, 3, point, 00 each. If the concession stand brought in dollar sign, 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

**slope-intercept form**of a linear function, start color #ca337c, y, equals, m, x, plus, b, end color #ca337c, where m and b are constants, 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.

#### Standard form

The

**standard form**of a linear function, start color #ca337c, A, y, plus, B, x, equals, C, end color #ca337c, 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!

## Your turn!

## Things to remember

The

**slope-intercept form**of a linear equation, y, equals, m, x, plus, 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

## Want to join the conversation?

- terrible tractor investment(101 votes)
- ill be doing on June 3rd,God bless us all(28 votes)
- me too brother, amen(8 votes)

- i can't understand this lesson , i tried for 5 days and tried in different moods , any advice ?(6 votes)
- think differently //(11 votes)

- Korean kids do this when they're 12. WE CAN DO THIS GUYS! YOU CAN DO THIS!(13 votes)
- It's easy, I don't have a problem with algebraic modifiers, but when it comes to the verbal part, I struggle because English is not my mother tongue, and that's a bit frustrating, but I'm still trying.(13 votes)
- dont be afraid from anything god is with us everything will be good(13 votes)
- Anyone for May 6th?(12 votes)
- Am I stupid or is this the exact same lesson as the last one?(7 votes)
- yeah it's like literally the same thing...(1 vote)

- Yo what up homies! Who else chilling in class(6 votes)
- Can we find out y intercept from just two points of the straight line? I mean without the knowing the value of Y(1 vote)
- Yep 100%. If you have 2 points of a straight line, first find the gradient using the gradient formula. Then input the x and y value of one of the points alongside the gradient value in y=mx+b than solve for b.(8 votes)