Find two pairs of coordinates for each equation by making a T-chart. Use the coordinates to graph the lines and find the solution. x + 2y = 6 2x - 3y = 12

To solve the system of equations by graphing, we'll first create T-charts for each equation, find pairs of coordinates, and then use these to graph the lines.

### Step 1: Solve for y in each equation.

1. **First equation: \( x + 2y = 6 \)**

   Solve for \( y \):

   \[
   2y = 6 - x
   \]
   \[
   y = \frac{6 - x}{2}
   \]

2. **Second equation: \( 2x - 3y = 12 \)**

   Solve for \( y \):

   \[
   -3y = 12 - 2x
   \]
   \[
   y = \frac{2x - 12}{3}
   \]

### Step 2: Create T-Charts and Find Coordinates

1. **For \( x + 2y = 6 \):**

   Choose values for \( x \) and find corresponding \( y \) values.

   | \( x \) | \( y \) | Coordinates |
   |:------:|:------:|:-----------:|
   |   0    |   3    |   (0, 3)    |
   |   2    |   2    |   (2, 2)    |
   |   4    |   1    |   (4, 1)    |

2. **For \( 2x - 3y = 12 \):**

   Choose values for \( x \) and find corresponding \( y \) values.

   | \( x \) | \( y \) | Coordinates |
   |:------:|:------:|:-----------:|
   |   0    |  -4    |   (0, -4)   |
   |   3    |  -2    |   (3, -2)   |
   |   6    |   0    |   (6, 0)    |

### Step 3: Graph the Lines and Find the Solution

- Plot the coordinates from both T-charts on a graph.
- Draw lines through the points for each equation.
- The intersection point of these two lines will be the solution to the system.

If you graph these correctly, you will find that the lines intersect at the point \( (6, 0) \), which is the solution to the system of equations.

### Solution:
The solution is \( (6, 0) \).

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Do you have any questions, or would you like further details on any part of this process?

### Related Questions:

1. What happens if two lines do not intersect?
2. How can we verify the solution using substitution?
3. Can the solution be a fraction or decimal?
4. What if the lines overlap completely?
5. What method can be used if graphing is not practical?

**Tip:** Always double-check your graphs by substituting the intersection point back into both original equations!

Learn how to graph linear equations x + 2y = 6 and 2x - 3y = 12 using T-charts and find their intersection point. This problem covers fundamental concepts in algebra, suitable for Grades 7-9 students.

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