Question

a system of linear equations is shown on the graph. the equation of one of the lines is $y = -\frac{1}{2}x + 4$. what is the equation of the other line in the system?

Explanation:

Step1: Identify the y-intercept of the other line

From the graph, the other line crosses the y - axis at \(y=-1\), so the y - intercept \(b=-1\).

Step2: Calculate the slope of the other line

We can use two points on the line. Let's take the y - intercept \((0, - 1)\) and another point, say \((1,2)\) (by looking at the graph, we can see that when \(x = 1\), \(y=2\)). The slope formula is \(m=\frac{y_2 - y_1}{x_2 - x_1}\).
Substituting \((x_1,y_1)=(0,-1)\) and \((x_2,y_2)=(1,2)\) into the slope formula:
\(m=\frac{2-(-1)}{1 - 0}=\frac{2 + 1}{1}=3\)

Step3: Write the equation of the line

The slope - intercept form of a line is \(y=mx + b\), where \(m\) is the slope and \(b\) is the y - intercept.
We found that \(m = 3\) and \(b=-1\), so the equation of the line is \(y = 3x-1\).

Answer:

\(y = 3x-1\)