Recognize and represent proportional relationships · lesson

The constant of proportionality

💡 In y = kx, the constant k is y divided by x, and it is the same for every pair.

When y is proportional to x, there is one number k so that y = kx. The k is called the constant of proportionality.

To find k, divide y by x for any pair. Every pair gives the same k.

🧠 Let's work it out

y is proportional to x, and y = 35 when x = 5. Find k.

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💡 Once you know k, multiply any x by k to find its y.

The equation y = kx is a rule. Put in an x, multiply by k, and get y.

If you don't know k yet, find it first from the pair you are given.

🧠 Let's work it out

y = 2.5x. What is y when x = 6?

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💡 Find the amount for one, then multiply to get the amount you need.

Many word problems are proportional: price per item, miles per gallon, kilometers per centimeter on a map.

Find the unit rate (the amount for one). Then multiply by the new amount. Or divide to find how many you need.

🧠 Let's work it out

5 tickets cost $40. How much do 12 tickets cost?

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💡 A relationship is proportional only if y ÷ x is the same for every pair, so its equation looks like y = kx with nothing added.

To test a table, divide y by x for every pair. If you always get the same number, it is proportional. If not, it isn't.

To test an equation, look for the form y = kx. If a number is added, like y = 4x + 2, it is not proportional. Its graph does not go through (0, 0).

🧠 Let's work it out

Table: x is 2, 3, 4 and y is 6, 9, 13. Is y proportional to x?

What do you think comes next?