Question

3. Is the relationship shown by the data linear? If so, model the data with an equation X | Y 1 ,5| 5,10| 9,15| 13,20|

Answer 1

It is linear.

The equation is 5x - 4y = -15.

Explanation:

If it is linear then the slope between consecutive points will be the same.

Slope = (10-5)/(5-1) = 5/4.

slope = (15-10)/ (9-5) = 5/4

slope = (20-15)/ (13-9) = 5/4.

So the data relationship is linear.

y - y1 = m(x - x1)

Using the point (1, 5)

y - 5 = 5/4(x - 1)

y = 5/4x - 5/4 + 5

y = 5/4x + 15/4

4y = 5x + 15

5x - 4y = -15.

Answer 2

`y=(5)/(4)x+(15)/(4)`

Explanation:

x | y

-------

1 5

5 10

9 15

13 20

This one is linear because as x goes up by the same number so does y. So the ratio of difference of y to difference of x is the same per pair of points.

So a linear equation in slope-intercept form is y=mx+b where m is the slope and b is the y-intercept.

To find the slope, I'm going to line up the points vertically and subtract, then put 2nd difference over 1st difference. Like so,

( 1 , 5)

-( 5 , 10)

----------------

-4 -5

So the slope is 5/4 which makes sense since the y's are going up by 5 each time and the x's are going up by 4 each time.

So we have m=5/4. Let's plug that into our y=mx+b.

y=5/4 x+b

To find b, we need to use y=5/4 x+b along with one of the given points.

Choose; it doesn't matter. I like (1,5) I guess.

y=5/4 x +b with (1,5)

5=5/4 (1)+b

5=5/4 +b

Subtract 5/4 on both sides:

5-5/4=b

20/4-5/4=b (Found a common denominator)

15/4=b

The y-intercept is 15/4 so b=15/4.

So the equation for the line in slope-intercept form is y=5/4 x +15/4.

`y=(5)/(4)x+(15)/(4)`