Determine If The Tables And Graphs Below Express A Direct Variation…

Determine if the tables and graphs below express a direct variation between the variables If they do, find the constant of variation and an equation that defines the relation. 1. 2. 3. 4. 5. 6. X y X y X y X y X y X y 1 -3 7 14 -21 -28 3 6₁ -9 -12 -15 10 -20 25 -3 2 -4 5 2 1 2 3 4 -6 -9 -12 16 12 6 7 3 4 2 3 20 15 8 9 5 4 24 28 18 21 10 11 12 13 32 24 82​

Answer:

To determine if the tables and graphs express a direct variation between the variables, we need to check if the ratio of y-values to x-values remains constant. If the ratio is constant, then there is a direct variation between the variables.

Let’s analyze each set of values:

1. X y

1 -3

2 -4

The ratio of y-values to x-values is (-3)/1 = -4/2 = -2. Since the ratio is constant, there is a direct variation. The constant of variation can be found by taking any x-y pair, for example, (1, -3), and dividing y by x: -3/1 = -3. Therefore, the equation that defines the relation is y = -3x.

2. X y

7 14

5 10

The ratio of y-values to x-values is 14/7 = 10/5 = 2. The ratio is constant, indicating a direct variation. The constant of variation is found by taking any x-y pair, such as (7, 14), and dividing y by x: 14/7 = 2. Therefore, the equation that defines the relation is y = 2x.

3. X y

-21 -28

-15 -20

The ratio of y-values to x-values is (-28)/(-21) = (-20)/(-15) = 4/3. The ratio is constant, indicating a direct variation. The constant of variation is found by taking any x-y pair, such as (-21, -28), and dividing y by x: -28/(-21) = 4/3. Therefore, the equation that defines the relation is y = (4/3)x.

4. X y

-9 -12

-6 -9

The ratio of y-values to x-values is (-12)/(-9) = (-9)/(-6) = 4/3. The ratio is constant, indicating a direct variation. The constant of variation is found by taking any x-y pair, such as (-9, -12), and dividing y by x: -12/(-9) = 4/3. Therefore, the equation that defines the relation is y = (4/3)x.

5. X y

-20 -3

-12 -6

The ratio of y-values to x-values is (-3)/(-20) = (-6)/(-12) = 1/4. The ratio is not constant, indicating that there is no direct variation between the variables.

6. X y

-15 -9

-10 -6

The ratio of y-values to x-values is (-9)/(-15) = (-6)/(-10) = 3/5. The ratio is not constant, indicating no direct variation between the variables.

In summary, the tables and graphs that express direct variation are 1 and 2. The equations that define the relations are:

1. y = -3x

2. y = 2x

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