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# Understanding linear relationships | Lesson

This lesson covers the following OSP skills: Linear equation word problems Linear function word problems Interpreting linear functions

## What are linear relationships, and how frequently do they appear on the test?

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.
Linear relationships appear frequently on the SAT: about 25, percent of the SAT Math test involves linear relationships! For this reason, we recommend you get comfortable working with linear relationships in their many forms.
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
On your official SAT, you'll likely see 5 to 7 questions like those found in this lesson. The skills covered here will also be important for the following lessons:
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, equals, x
By plugging numbers into the equation, we can find some relative values of x and y.
xy
00
11
22
33
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

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 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 (frequent)
• Write our own equation and then solve it (frequent)
• Solve a given equation based on the word problem (infrequent)

### 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, equals, 35, plus, 16, t.

## 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, 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!

Try: build a linear function
Shipping Charges
Merchandise weight (pounds)Shipping charge
5dollar sign, 16, point, 49
10dollar sign, 23, point, 99
25dollar sign, 46, point, 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 dollar sign, 600, which she will pay for with an initial payment of dollar sign, 150 and monthly installments of dollar sign, 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, point, 10, x, plus, 0, point, 20, y, equals, 0, point, 12, left parenthesis, x, plus, y, right parenthesis
Lawrence will mix x milliliters of a 10, percent by mass saline solution with y milliliters of a 20, percent by mass saline solution in order to create a 12, percent by mass saline solution. The equation above represents this situation. If Lawrence uses 100 milliliters of the 20, percent by mass saline solution, how many milliliters of the 10, percent by mass saline solution must he use?

Practice: interpret a Linear function
y, equals, 35, x, plus, 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 x, y-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 dollar sign, 330, comma, 000. The value of the machine depreciates by the same amount each year so that after 10 years the value will be dollar sign, 80, comma, 000. Which of the following equations gives the value, v, of the harvester, in dollars, t years after it was purchased for 0, is less than or equal to, t, is less than or equal to, 10 ?

## Things to remember

start text, s, l, o, p, e, end text, equals, start fraction, start text, c, h, a, n, g, e, space, i, n, space, end text, y, divided by, start text, c, h, a, n, g, e, space, i, n, space, end text, x, end fraction, equals, start fraction, y, start subscript, 2, end subscript, minus, y, start subscript, 1, end subscript, divided by, x, start subscript, 2, end subscript, minus, x, start subscript, 1, end subscript, end fraction
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?

• I would like to make a request, could you please add a video that explains what exactly is the slope-intercept form?
• Hello
I have just started studying SAT Math Test lessons and it was my first lesson (Understanding liner relationships)
-How can I more practice the mentioened lesson?

This is very important to say I finished my High-School around 10 years ago and there is a huge gap! So If I study and finish all 40 lessons in SAT Math Test lesson part that you have mentioned them will be enough to get ready for SAT Test Math part?
• Hello,
Khan Academy also has a bunch of SAT practice (the skill practices should cover the same topics as the videos) and full-length practice tests. If you take a diagnostic math quiz or a full test, Khan Academy will recommend certain practices and help you focus on the areas you need to work on... which is probably easier than studying by going through every lesson. :) I hope this helps. Good luck on the SAT!
• I have a confusion in the last question. Why we are choosing 0 as y2 and 10 as y1? Why can`t we reverse them?
What I mean to say is, if we create a graph for the values then it would be a negative graph(with a negative slope).So 330,000 should be x1 and vice versa.
can anyone explain this?
• You very well could reverse them, and it should give you the same slope. The slope formula doesn't have to use an earlier point on the line and a later point. Instead, it is just any two points.
Typically, you would have the earlier time as t1 and the later time as t2 because that makes chronological sense, but this is the SAT, so you if you can cut out a step or two in the interest of saving time, that's good.
(330,000 - 80,000) / (0 - 10) is at least to me, visually a lot easier to do than (80,000 - 330,000) / (10 - 0) because negatives with big numbers can be scary if you're in a hurry.
It's a matter of personal preference whether you want to always subtract the latter data point from the former or mix it up to fit the situation, do whichever one feels more natural to you.
• I am confused about the formula taught in algebra 1 for linear relationships with "n" in it.
• i need to know the easiest way to solve linear equations i'm really slow with solving them
• could you make a video that explains how two equations can have no solution?
(1 vote)
• You might find these resources on Khan Academy helpful: article "Number of solutions to system of equations review" and video "Systems of equations number of solutions: fruit prices (1 of 2)".

Just put those titles (or anything you want more info on) into the search bar at the top of Khan's webpages, and you can find some useful things!

If you can't find the material you want by searching there, make sure to put any content requests in the Support Community. The link to it's at the bottom of the page!
• The equation above models yyy, the amount in dollars charged by a website hosting service to host a website for mmm months. The total cost consists of a one-time setup fee plus a monthly charge for hosting. When the equation is graphed in the xyxyx, y-plane, what does the yyy-intercept of the graph represent in terms of the model?
Choos
• How are you? I hope you're okay