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### Course: Digital SAT Math>Unit 2

Lesson 2: Linear equation 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 $xy$-plane. Linear relationships are very common in everyday life.
In this lesson, we'll:
1. Review the basics of linear relationships
2. Practice writing linear equations based on word problems
3. 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 $xy$-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

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 $\mathrm{}17,000$ is to be purchased with an initial payment of $\mathrm{}5,000$ 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!

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 $179=35+16t$.

## What are important features of linear functions?

### Linear equations in slope-intercept form

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, $y=mx+b$, where $m$ and $b$ are constants, tells us both the slope and the $y$-intercept of the line:
• The slope is equal to $m$.
• The $y$-intercept is equal to $b$.

#### Standard form

The standard form of a linear function, $Ay+Bx=C$, 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 $y$.

#### 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$$\mathrm{}16.49$
$10$$\mathrm{}23.99$
$25$$\mathrm{}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:

Practice: write a linear equation
Tamika purchases a new mattress for $\mathrm{}600$, which she will pay for with an initial payment of $\mathrm{}150$ and monthly installments of $\mathrm{}30$. Which of the following equations can be used to find the number of monthly installments, $m$, required to complete the purchase, assuming there are no taxes or fees?

Practice: solve a linear equation
$0.10x+0.20y=0.12\left(x+y\right)$
Lawrence will mix $x$ milliliters of a $10\mathrm{%}$ by mass saline solution with $y$ milliliters of a $20\mathrm{%}$ by mass saline solution in order to create a $12\mathrm{%}$ by mass saline solution. The equation above represents this situation. If Lawrence uses $100$ milliliters of the $20\mathrm{%}$ by mass saline solution, how many milliliters of the $10\mathrm{%}$ by mass saline solution must he use?

Practice: interpret a Linear function
$y=35x+550$
The equation above models $y$, the amount in dollars charged by a website hosting service to host a website for $m$ months. The total cost consists of a one-time setup fee plus a monthly charge for hosting. When the equation is graphed in the $xy$-plane, what does the $y$-intercept of the graph represent in terms of the model?

Practice: Linear function word problems
A farm purchased a combine harvester valued at $\mathrm{}330,000$. The value of the machine depreciates by the same amount each year so that after $10$ years the value will be $\mathrm{}80,000$. Which of the following equations gives the value, $v$, of the harvester, in dollars, $t$ years after it was purchased for $0\le t\le 10$ ?

## Things to remember

The slope-intercept form of a linear equation, $y=mx+b$, 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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