The problem involves solving and graphing a system of linear equations. Let's break it into steps:

Step 1: Understand the equations

The system provided is:

1. $3x + y = -4$
2. $-5x + y = 8$

These equations are rewritten in slope-intercept form ($y = mx + b$):

1. $y = -3x - 4$
2. $y = 5x + 8$

Step 2: Graph the equations

For graphing, you find the y-intercepts ($b$) and slopes ($m$):

1. Equation $y = -3x - 4$:

   • Slope ($m$): $-3$
   • Y-intercept ($b$): $-4$

   Start at $(0, -4)$ on the graph and use the slope $-3$ (down 3 units, right 1 unit) to plot additional points.

2. Equation $y = 5x + 8$:

   • Slope ($m$): $5$
   • Y-intercept ($b$): $8$

   Start at $(0, 8)$ and use the slope $5$ (up 5 units, right 1 unit) to plot additional points.

These lines should intersect, and that intersection point represents the solution to the system of equations.

Step 3: Use the graph to estimate the solution

From the image, it appears that the two lines intersect approximately at $(-2, 2)$. This can be verified by substituting $(-2, 2)$ into both original equations.

Verify:

1. Substitute into $3x + y = -4$:
   $3(-2) + 2 = -6 + 2 = -4$ (True)

2. Substitute into $-5x + y = 8$:
   $-5(-2) + 2 = 10 + 2 = 8$ (True)

Both equations are satisfied, so the solution is indeed $(-2, 2)$.

Final Explanation:

The solution $(-2, 2)$ was determined by identifying the point where the two lines intersected on the graph. This method is valid because the intersection of two lines represents the values of $x$ and $y$ that satisfy both equations simultaneously.

Do you need further clarification or want me to graph this digitally? Here are 5 related questions for deeper understanding:

1. How do you determine the slope and y-intercept from the standard form of a linear equation?
2. What are other methods for solving systems of equations besides graphing?
3. How can you verify the solution algebraically without graphing?
4. What happens when two lines are parallel or coincide in a system of equations?
5. How does changing the slope or intercept affect the graph of a line?

Tip: Always verify your graphing solution algebraically to ensure precision!