Using direct variation

Consider the three points (-4,-3) , (20,15) , and (48,36).

Part A
Which points are on the same line that passes through (-4,-3) , (20,15) , and (48,36)?
Select all that apply.

A.)  (-8,-6)     B.)  (-2,-1)     C.)  (0,0)
D.)  (4,3)       E.)  (6,8)

Part C
Do the points on the line y = 3x - 2 have a constant ratio of the y-coordinate to the
x-coordinate for any point on the line except for the y-intercept?  Explain your answer.

Part B
Use the information from Part A to explain why the ratio of the y-coordinate to the x-coordinate is the same for any point on the line except the y-intercept.

Explain why this is not true for the y-intercept.

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Part A
The model for direct variation is y = kx

k = y/x   {divided each side by x}

If a point is on the same line, it should have
the same y/x ratio, equaling the constant of variation, k.

For the first point (-4,-3), the constant of variation is:
k = (-3) / (-4) = 3/4   {reduced the constant of variation}

The points that have a y/x ratio of 3/4 are:
A.)  (-8,-6)
and
D.)  (4,3)

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Part B
In part A, the constant of variation, k = y/x , was found to be 3/4 .

Every point on the line has a ratio of the y-coordinates to the x-coordinates being 3/4,
except the y-intercept.

At the y-intercept, the value of x is 0.  Therefore, if x is zero, then k=y/x would be undefined.

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Part C
No.  The equation y = 3x – 2 is not in the form of direct variation, which is y = kx, because 
of the constant term, -2.

Example:
If x = 2
y = 3(2) – 2   {substituted 2 for x}
y = 6 – 2   {multiplied}
y = 4   {subtracted}
(2 , 4)   {the coordinates on the line}

The ratio of the y-coordinate to the x-coordinate would be 4/2 = 2  

Example:
If x = 5  
y = 3(5) – 2   {substituted 5 for x}
y = 15 – 2   {multiplied}
y = 13   {subtracted}
(5 , 13)   {the coordinates on the line}

The ratio of the y-coordinate to the x-coordinate would be 13/5. 

Therefore, the ratio of the y-coordinate to the x-coordinate is not constant.

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