Question

write the equation of the line in fully simplified slope - intercept form.

Explanation:

Step1: Identify slope-intercept form

The slope - intercept form of a line is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y - intercept.

Step2: Find the y - intercept (\(b\))

The line crosses the y - axis at \((0,-3)\), so \(b=-3\).

Step3: Calculate the slope (\(m\))

We can use two points on the line. Let's take \((0, - 3)\) and \((3, - 1)\) (we can also use other points like \((6,1)\) etc.). The slope formula is \(m=\frac{y_2 - y_1}{x_2 - x_1}\).
Using \((x_1,y_1)=(0,-3)\) and \((x_2,y_2)=(3,-1)\), we have \(m=\frac{-1-(-3)}{3 - 0}=\frac{-1 + 3}{3}=\frac{2}{3}\).

Step4: Write the equation

Substitute \(m = \frac{2}{3}\) and \(b=-3\) into the slope - intercept form \(y=mx + b\).
We get \(y=\frac{2}{3}x-3\).

Answer:

\(y=\frac{2}{3}x - 3\)