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Write Equations for Proportional Relationships

Students represent proportional relationships with equations of the form y = kx, where k is the constant of proportionality. They identify k from a table or graph, write the equation, and use it to find missing values.

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Write Equations for Proportional Relationships

Grade 7 · Ratios and Proportional Relationships · question 1 of 3

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20 rungs. A star at the top.

(6, 12) on y = kx: k =

How it is taught

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A proportional equation is y equals k times x.

k is y divided by x at a point.

Read it: The constant of proportionality

In this table every y is 5 times its x: 5 ÷ 1 = 5, 10 ÷ 2 = 5 and 30 ÷ 6 = 5. That one number is k, the constant of proportionality, so y = 5x.

  1. Divide each y by its x.
  2. If every quotient is the same number, that number is k.
  3. Write the equation y = kx with that k.

k is the y that goes with one x: here 1 goes with 5.

Every x is multiplied by the same k, so 6 goes with 6 × 5 = 30.

Knowing k finds any y from its x, and any x from its y: x = y ÷ k.

k = y ÷ x, the same for every pair, and y = kx.

Dividing x by y: 6 ÷ 30.

k is how much y for one x: y ÷ x = 30 ÷ 6 = 5.

(9, 63) on y = kx: k = ___ — 7. k is the y for one x: 63 ÷ 9 = 7, so y = 7x.

y = 40x for a steady-speed car

A car drives at a steady speed. Let's find an equation that predicts how far it travels.

Here's a table. After one hour: forty miles. After two hours: eighty. After three hours: one hundred twenty.

Let's check the ratio of miles to hours. Forty divided by one is forty.

Eighty divided by two is also forty.

And one hundred twenty divided by three is forty too!

The ratio is the same every row! That special number has a name — k, the constant of proportionality. Here, k equals forty.

A car drives at a steady speedHow can we predict the distance for any time?Hours (x)Miles (y)140280312040 ÷ 1 = 4080 ÷ 2 = 40120 ÷ 3 = 40Same ratio every time → k = 40k is the constant of proportionalityy = k · xthe rule for every proportional relationshipand we know k = 40y = 40xIf x = 5 hours, what is y?y = 40 × 5y = 200ANSWER200miles in 5 hours
Grade 7 · A car drives 40 miles every hour

Every proportional relationship follows the same rule: y equals k times x.

So we plug in our k. Our equation is y equals forty x.

Now let's use it. How far does the car go in five hours?

Substitute x with five: y equals forty times five.

Forty times five is two hundred.

Our equation y equals forty x tells us the car travels two hundred miles in five hours. Awesome work!

What you see

While they climb, your page writes one true line to ask about tonight, and a short note each Sunday.

MayaGrade 3 · Mathematics
Did you know?

A chessboard has 64 squares — share them out between the 8 rows and each row holds exactly 8.

Today’s goal
1 star — one skill mastered, every rung of its ladder climbed
Climbing now
Divide within 100

An example student.

MayaGrade 3 · Mathematics

Maya earned 2 stars this week — Interpret multiplication as equal groups, which took 6 answers over 3 days before its star landed 3 days ago and Multiply by 6, 7, 8 and 9. Next up: Divide within 100.

An example week.

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