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A relationship is said to have direct variation when one variable changes and the second variable changes proportionally; the ratio of the second variable to the first variable remains constant. For example, when y varies directly as x, there is a constant, k, that is the ratio of y:x.

The table below represents a situation where Chloe wants to save money to purchase a tablet computer. She decides to save \$40 each week.

 Chloe's Savings Week (w) Amount Saved (y) 1 \$40 2 \$80 3 \$120 4 \$160 5 \$200

For situations that are linear and proportional, the two variables are said to vary directly with one another. In other words, for Chloe’s savings plan, we can say the following: The equation for a linear direct variation is y = kx, where k is the slope of the line y = mx + b, and the y-intercept, or b, equals zero.  Since the direct variation's equation is the same as the slope intercept equation of a line, the linear equation can be used to solve direct variation as shown in the video below.*

Source: Direct Variation, bullcleo1, You Tube Direct variations are very useful in the real world. The video below shows several examples of real-world applications of direct variation.

Source: Direct Variation Models, Khan Academy, You Tube

### Practice

1. If y varies directly as x and x = 12 when y = 9, find the value of k, and then find the value of y when x = 32.

Interactive popup. Assistance may be required.

Substitute the known values for x and y into the equation y = kx, and then solve that equation for k. Use that value of k to calculate the value of y when x = 32. Interactive popup. Assistance may be required.  2. Sam earns \$8 per hour. If he works four hours, he earns \$32. Find k, and then find the number of hours that Sam would have to work to earn \$248.

Interactive popup. Assistance may be required.

What is the rate of change, or slope, in this situation? Interactive popup. Assistance may be required.

Because the rate of change, or slope, is \$8 per hour, k = 8.

y = 8x
248 over 8 248 8 = 8x over 8 8x 8
31 = x

Sam must work 31 hours to earn \$248. 3. *Some materials in this resource are from outside sources. TEA has chosen to include them for the value of the content even if grammatical or punctuation errors exist.