The task is to solve the system of linear equations by graphing. Let's break it down step by step based on the problem:

Given Equations:

1. $x - 3y = 6$
2. $4x + 3y = -6$

Steps to Solve:

Step 1: Rewrite each equation in slope-intercept form ($y = mx + b$).

Equation 1: $x - 3y = 6$ $-3y = -x + 6$ $y = \frac{1}{3}x - 2$

Equation 2: $4x + 3y = -6$ $3y = -4x - 6$ $y = -\frac{4}{3}x - 2$

Step 2: Graph each equation.

• For $y = \frac{1}{3}x - 2$:

  • The y-intercept is $-2$, so plot the point $(0, -2)$.
  • The slope is $\frac{1}{3}$, meaning for every 1 unit increase in $x$, $y$ increases by $1/3$.
  • Plot additional points like $(3, -1)$, $(6, 0)$, etc., to make the line clearer.

• For $y = -\frac{4}{3}x - 2$:

  • The y-intercept is also $-2$, so plot the point $(0, -2)$.
  • The slope is $-\frac{4}{3}$, meaning for every 3 units increase in $x$, $y$ decreases by 4.
  • Plot additional points like $(3, -6)$, $(-3, 2)$, etc.

Step 3: Identify the intersection point.

The intersection point represents the solution to the system of equations. By observing the graph:

• The two lines intersect at $(-3, -3)$.

Final Answer:

The solution to the system is: $\boxed{(-3, -3)}$

Would you like further clarification or additional examples of solving systems of equations?

Related Questions:

1. What are other methods to solve a system of linear equations besides graphing?
2. How can you check if a point lies on both lines of the system?
3. What happens if two lines are parallel when solving a system of equations?
4. How do you graph a line in standard form without converting to slope-intercept form?
5. How can you use graphing tools on a computer to visualize systems of equations more accurately?

Tip: When graphing lines, always double-check your calculations for the slope and y-intercept to ensure precision.