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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. \( 4x + 3y = -6 \)

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### 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: \(4x + 3y = -6\)**
\[
3y = -4x - 6
\]
\[
y = -\frac{4}{3}x - 2
\]

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#### 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?  

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### 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.

Solve the system of linear equations by graphing: x - 3y = 6 and 4x + 3y = -6.

Learn how to solve the system of linear equations x - 3y = 6 and 4x + 3y = -6 by graphing. Step-by-step instructions include converting equations to slope-intercept form, plotting points, and finding the intersection. Suitable for Grade 8-10 students.

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