Question

graph the line that passes through the points (-9, 3) and (6, 8) and determine the equation of the line.

Explanation:

Step1: Calculate the slope

The slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by \( m=\frac{y_2 - y_1}{x_2 - x_1} \). For the points \((-9, 3)\) and \((6, 8)\), we have \( x_1=-9,y_1 = 3,x_2=6,y_2=8 \). So \( m=\frac{8 - 3}{6-(-9)}=\frac{5}{15}=\frac{1}{3} \).

Step2: Use point - slope form

The point - slope form of a line is \( y - y_1=m(x - x_1) \). Let's use the point \((6, 8)\) (we could also use \((-9,3)\)). Substituting \( m = \frac{1}{3}\), \( x_1 = 6\) and \( y_1=8 \) into the point - slope form, we get \( y - 8=\frac{1}{3}(x - 6) \).

Step3: Simplify to slope - intercept form

Expand the right - hand side: \( y - 8=\frac{1}{3}x-2 \). Then add 8 to both sides: \( y=\frac{1}{3}x + 6 \).
To graph the line:

1. Plot the points \((-9,3)\) and \((6,8)\) on the coordinate plane.
2. Draw a straight line passing through these two points. The line should have a slope of \(\frac{1}{3}\) (for every 3 units we move to the right along the x - axis, we move up 1 unit along the y - axis) and a y - intercept of 6 (it crosses the y - axis at \((0,6)\)).

Answer:

The equation of the line is \( y=\frac{1}{3}x + 6 \), and the graph is a straight line passing through \((-9,3)\) and \((6,8)\) (with slope \(\frac{1}{3}\) and y - intercept 6).