Identify the Linear Equation with a Slope of -1 from the Graph

To match the graph to the correct equation, we will analyze the slope and y-intercept:

• Step 1: Identify y-intercept.
  From the graph, observe that the line crosses the y-axis at $y = 4$. Therefore, the y-intercept $b = 4$.

• Step 2: Calculate slope.
  Identify two points on the graph line, such as $(0, 4)$ and $(5, 0)$. Calculate slope using $m = \frac{\Delta y}{\Delta x}$:
  $m = \frac{0 - 4}{5 - 0} = \frac{-4}{5}$.

• **Step 3: Match to equations**.
  Now we have $m = -\frac{4}{5}$ and $b = 4$, so the equation should be $y = -\frac{4}{5}x + 4$.

After comparing with the choices, we see that choice $(2)$ $y = -\frac{4}{5}x + 4$ correctly matches the information derived from the graph.

Therefore, the equation that corresponds to the graph is $y = -\frac{4}{5}x + 4$.