Question

Is the relationship shown by the data linear? If so, model the data with an equation. x y 1 –4 7 –7 13 –10 19 –13

Answer 1

You can see that there is a proportional relationship between X and Y because every 6 of X decreases 3 in Y. In that way you can say is a linear relationship and you can construct a linear equation: Now, to calculate the equation you use: (Y - y1) = y2 - y1 (X - x1) x2 - x1 Let's peek two pairs (1,-4) and (7,-7) (Y - (-4)) = (-7 - (-4)) (X - 1) (7 - 1) Y + 4 = -3 (X - 1) 6 If you want to express the equation as aX + bY + C = 0 Y + 1/2 X + 7/2 = 0 If you want to express the equation as Y= ax + b Y = -1/2X - 7/2

Answer 2

Yes, the relationship shown by the data linear.

The equation is given by:

`y=(-1)/(2)x-(7)/(2)`

Explanation:

The table of values is given by:

x y

1 -4

7 -7

13 -10

19 -13

We know that the difference in each of the x-value is: 6

( Since, 7-1=6

13-7=6

19-13=6)

and if the difference in the each of the y-value is same then the relationship is linear.

( Since, a table of values represent a linear relationship if the rate of change is constant.

i.e. the ratio of change in y-values to the change in x-values)

Hence, we find the difference in y-value:

-7-(-4)= -3

-10-(-7)= -3

-13-(-10)= -3

Since, the difference in y-value is constant.

Hence, the relationship is linear.

Also, we know that the equation of the line will pass through (1,-4) and (7,-7)

Hence, the equation of line is calculated by:

`y-(-4)=(-7-(-4))/(7-1)* (x-1)\\\\i.e.\\\\y+4=(-3)/(6)* (x-1)\\\\i.e.\\\\y+4=(-1)/(2)* (x-1)\\\\i.e.\\\\y+4=(-1)/(2)x+(1)/(2)\\\\i.e.\\\\y=(-1)/(2)x+(1)/(2)-4\\\\i.e.\\\\y=(-1)/(2)x-(7)/(2)`