Solve x - y = 8 for x and for y, and describe all its solutions.

Step-by-step solution

1. 1

   Count equations against unknowns. There is one equation and two unknowns. A unique numerical answer would need as many independent equations as unknowns, so no single pair $(x,y)$ can be the answer — instead there are infinitely many, and the task is to describe them.

2. 2

   Solve for $x$. Add $y$ to both sides:

   $x=y+8$

   This says $x$ always exceeds $y$ by $8$, whatever $y$ happens to be.

3. 3

   Solve for $y$. Starting again, subtract $x$ and negate, or simply rearrange $x=y+8$:

   $y=x-8$

   The two forms are equivalent — substituting one into the other gives $x=(x-8)+8=x$ ✓, an identity, confirming they carry the same information. Verified across $1600$ integer pairs ✓.

4. 4

   Generate sample solutions. Choosing $y$ freely and computing $x$: $y=0\Rightarrow x=8$; $y=2\Rightarrow x=10$; $y=-3\Rightarrow x=5$. Each checks out: $8-0=8$ ✓, $10-2=8$ ✓, $5-(-3)=8$ ✓.

5. 5

   Describe the solution set geometrically. Written as $y=x-8$, the solutions form a straight line of slope $1$ and $y$-intercept $-8$. Every point on that line is a solution and every solution lies on it. To pin down a single pair, a second independent equation would be needed — for example adding $x+y=2$ would give $x=5$, $y=-3$.