Understanding linear relationships | Lesson (article) | Khan Academy (2024)

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.

Maya is 3 inches taller than Geoff.

We can express the linear relationship between Maya's height in inches, y, and Geoff's heights in inches, x:

Maya’s height=Geoff’s height+3y=x+3

Tai runs 2 miles every day.

We can express the linear relationship between the distance Tai runs in miles, y, and the number of days they run, x.

distance=2daysy=2x

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.

xy
00
11
22
33

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

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Modeling with linear equations: gym membership & lemonade

See 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 $17,000 is to be purchased with an initial payment of $5,000 and monthly payments of $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?

We're given three important values here: 17,000, 5,000, and 240.

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

The total price, $17,000, equals the sum of the other payments: the initial $5,000 payment and the $240 paid each month (m).

17,000=5,000+240m

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 ?

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

P=2+2w=2(w+8)+2w=2w+16+2w=4w+16

The perimeter is equal to the sum of 16 and 4 times the width.

P=4w+16

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

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

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

total money=hot dog money+peanuts money196=3.00(42)+2.50(p)

The total revenue of $196 is equal to the sum of the revenue from hot dogs and the revenue from bags of peanuts. We can simplify and solve for p:

196=126+2.50p70=2.50p28=p

The concession stand sold 28 bags of peanuts.

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

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Constructing linear equations from context

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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.

    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 y by the change in x:

    slope=change inychange inx=y2y1x2x1

  • The y-intercept is equal to b.

    The y-intercept of a line is the y-value when x=0.

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

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

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$16.49
10$23.99
25$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:

Your turn!

Practice: write a linear equation

Tamika purchases a new mattress for $600, which she will pay for with an initial payment of $150 and monthly installments of $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?

Choose 1 answer:

Choose 1 answer:

  • 600=30m150

  • 600=30m

  • 600=150m+30

  • 600=30m+150

Practice: solve a linear equation

0.10x+0.20y=0.12(x+y)

Lawrence will mix x milliliters of a 10% by mass saline solution with y milliliters of a 20% by mass saline solution in order to create a 12% by mass saline solution. The equation above represents this situation. If Lawrence uses 100 milliliters of the 20% by mass saline solution, how many milliliters of the 10% by mass saline solution must he use?

Choose 1 answer:

Choose 1 answer:

  • 100

  • 200

  • 400

  • 600

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?

Choose 1 answer:

Choose 1 answer:

  • A setup fee of $550

  • A monthly charge of $35

  • A monthly charge of $550

  • Total monthly charges of $585

Practice: Linear function word problems

A farm purchased a combine harvester valued at $330,000. The value of the machine depreciates by the same amount each year so that after 10 years the value will be $80,000. Which of the following equations gives the value, v, of the harvester, in dollars, t years after it was purchased for 0t10 ?

Choose 1 answer:

Choose 1 answer:

  • v=80,00025,000t

  • v=330,00025,000t

  • v=330,000+25,000t

  • v=330,00080,000t

Things to remember

slope=change inychange inx=y2y1x2x1

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
Understanding linear relationships | Lesson (article) | Khan Academy (2024)
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