Q. Which is the graph of \( y – 3 = x + 6 \)

Problem: Write the equation \( y – 3 = x + 6 \) in slope-intercept form and find the slope, the x- and y-intercepts, and describe the graph.

Step 1 — Start with the given equation

\[
y – 3 = x + 6
\]

Step 2 — Isolate \(y\) to obtain slope-intercept form

Add \(3\) to both sides:

\[
y – 3 + 3 = x + 6 + 3
\]

which simplifies to

\[
y = x + 9
\]

This is in the form \(y = mx + b\).

Step 3 — Determine the slope

Compare \(y = x + 9\) with \(y = mx + b\). The coefficient of \(x\) is

\[
m = 1.
\]

Step 4 — Find the y-intercept

Set \(x = 0\):

\[
y = 0 + 9 = 9,
\]

so the y-intercept is the point \((0,\,9)\).

Step 5 — Find the x-intercept

Set \(y = 0\):

\[
0 = x + 9
\]

which gives

\[
x = -9,
\]

so the x-intercept is the point \((-9,\,0)\).

Step 6 — Describe the graph

The graph is a straight line with slope \(1\) passing through the points \((0,\,9)\) and \((-9,\,0)\).

Answer

\[
y = x + 9,\quad m = 1,\quad \text{x-intercept }(-9,\,0),\quad \text{y-intercept }(0,\,9)
\]