Sure! Let's solve this step-by-step:
### Part (a): Graph the line through the points (-6, 4) and (6, -4).
1. Identify the points: We have the points \(\((-6, 4)\)\) and \(\((6, -4)\)\).
2. Plot the points:
- Point \(\((-6, 4)\)\) is located 6 units to the left of the origin and 4 units up.
- Point \(\((6, -4)\)\) is located 6 units to the right of the origin and 4 units down.
3. Draw the line:
- Draw a straight line passing through both points. This line represents the linear relationship between \(\(x\)\) and \(\(y\)\) that goes through these points.
### Part (b): Write an equation of the line in the form \(\( y = mx + b \)\).
1. Calculate the slope \(\( m \)\):
The slope \(\( m \)\) of a line passing through the points \(\((x_1, y_1)\)\) and \(\((x_2, y_2)\)\) is given by:
\(\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]\)
For our points \(\((x_1, y_1) = (-6, 4)\)\) and \(\((x_2, y_2) = (6, -4)\)\):
\(\[
m = \frac{-4 - 4}{6 - (-6)} = \frac{-8}{6 + 6} = \frac{-8}{12} = -\frac{2}{3}
\]\)
2. Find the y-intercept \(\( b \)\):
The equation of the line is \(\( y = mx + b \)\). To find \(\( b \)\), use one of the points. We'll use \(\((-6, 4)\)\):
\(\[
y = mx + b \\
4 = \left(-\frac{2}{3}\right)(-6) + b \\
4 = 4 + b \\
b = 4 - 4 \\
b = 0
\]\)
3. Write the equation:
The equation of the line in slope-intercept form \(\( y = mx + b \)\) is:
\(\[
y = -\frac{2}{3}x + 0 \\
y = -\frac{2}{3}x
\]\)
So, the line through the points \(\((-6, 4)\)\) and \(\((6, -4)\)\) has the equation \(\( y = -\frac{2}{3}x \)\).
