Question

x y
-3 -3
0 -2
3 -1
6 0

Explanation:

Step1: Identify two points

Let's take the points \((0, -2)\) and \((3, -1)\) (or any two points from the table).

Step2: Calculate the slope \(m\)

The slope formula is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Using \((x_1,y_1)=(0, -2)\) and \((x_2,y_2)=(3, -1)\), we get \(m=\frac{-1 - (-2)}{3 - 0}=\frac{1}{3}\).

Step3: Determine the y - intercept \(b\)

Using the point - slope form \(y = mx + b\) and the point \((0, -2)\) (where \(x = 0\)), when \(x = 0\), \(y=b\). So \(b=-2\).

Step4: Write the equation of the line

The equation of a line in slope - intercept form is \(y=mx + b\). Substituting \(m = \frac{1}{3}\) and \(b=-2\), we get \(y=\frac{1}{3}x-2\). We can also check with other points. For \(x = 6\), \(y=\frac{1}{3}(6)-2=2 - 2=0\), which matches the table.

Answer:

The equation of the line represented by the table is \(y=\frac{1}{3}x - 2\) (If the question was to find the equation of the linear relationship. If it was something else, the process can be adjusted accordingly. For example, if it was to find the slope, the answer is \(\frac{1}{3}\); if it was to find the y - intercept, the answer is \(-2\)).