Certainly, let's go through the step-by-step solution to determine the equation of the line passing through the points (3, -12) and (9, -24), and identify any errors in your steps.
### Given:
Points on line B: [tex]\((3, -12)\)[/tex] and [tex]\((9, -24)\)[/tex]
### Step 1: Determine the Slope for B
The formula for the slope [tex]\(m\)[/tex] between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Substitute the given points:
[tex]\[ m = \frac{-24 - (-12)}{9 - 3} = \frac{-24 + 12}{9 - 3} = \frac{-12}{6} = -2 \][/tex]
So, the slope [tex]\(m\)[/tex] of line B is [tex]\(-2\)[/tex].
### Step 2: Determine the [tex]\(y\)[/tex]-intercept for B
The slope-intercept form of a line is:
[tex]\[ y = mx + b \][/tex]
where [tex]\(m\)[/tex] is the slope and [tex]\(b\)[/tex] is the [tex]\(y\)[/tex]-intercept.
Using one of the points, e.g., [tex]\((3, -12)\)[/tex]:
[tex]\[ y = mx + b \][/tex]
[tex]\[ -12 = -2(3) + b \][/tex]
[tex]\[ -12 = -6 + b \][/tex]
Solving for [tex]\(b\)[/tex]:
[tex]\[ b = -12 + 6 = -6 \][/tex]
So, the [tex]\(y\)[/tex]-intercept [tex]\(b\)[/tex] for line B is [tex]\(-6\)[/tex].
### Step 3: Write the Equation in Slope-Intercept Form
With [tex]\(m = -2\)[/tex] and [tex]\(b = -6\)[/tex], the equation of line B is:
[tex]\[ y = -2x - 6 \][/tex]
### Summary
The correct equation for line B, based on the coordinates [tex]\((3, -12)\)[/tex] and [tex]\((9, -24)\)[/tex], is:
[tex]\[ y = -2x - 6 \][/tex]
There was an error in your calculation of the [tex]\(y\)[/tex]-intercept in Step 2, and the final equation for line B should be corrected as shown above.
