Question

x | y
-9 | 12
-3 | 10
0 | 9
3 | 8
9 | 6
21 | 2

Explanation:

Step1: Calculate the slope between two points

Take two points \((x_1,y_1)=(-9,12)\) and \((x_2,y_2)=(-3,10)\). The slope formula is \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Substitute the values: \(m=\frac{10 - 12}{-3 - (-9)}=\frac{-2}{6}=-\frac{1}{3}\).

Step2: Check with another pair of points

Take \((x_1,y_1)=(-3,10)\) and \((x_2,y_2)=(0,9)\). Using the slope formula: \(m=\frac{9 - 10}{0 - (-3)}=\frac{-1}{3}=-\frac{1}{3}\). The slope is consistent, so the relationship is linear with slope \(-\frac{1}{3}\).

Step3: Find the equation of the line

Using the point - slope form \(y - y_1=m(x - x_1)\) with point \((0,9)\) (the y - intercept, since \(x = 0\) here), the equation is \(y-9=-\frac{1}{3}(x - 0)\), which simplifies to \(y=-\frac{1}{3}x + 9\). We can verify with other points. For \(x = 3\), \(y=-\frac{1}{3}(3)+9=-1 + 9 = 8\) (matches the table). For \(x = 9\), \(y=-\frac{1}{3}(9)+9=-3 + 9 = 6\) (matches the table). For \(x = 21\), \(y=-\frac{1}{3}(21)+9=-7 + 9 = 2\) (matches the table).

Answer:

The relationship between \(x\) and \(y\) is linear, and the equation of the line is \(y =-\frac{1}{3}x + 9\) (or we can state the slope is \(-\frac{1}{3}\) and the y - intercept is 9).