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 a T-chart for each equation to find two pairs of coordinates. Then, we'll use these coordinates to graph the lines and find their intersection, which represents the solution to the system.

### **Equation 1: \( x + 2y = 6 \)**

#### Step 1: Solve for \( y \)
To make it easier to fill in the T-chart, solve the equation for \( y \):
\[
2y = 6 - x
\]
\[
y = \frac{6 - x}{2}
\]

#### Step 2: Create the T-Chart
Let's choose two values for \( x \) and find the corresponding \( y \) values.

| \( x \) | \( y \) |
|:---:|:---:|
| 0 | \( y = \frac{6 - 0}{2} = 3 \) |
| 2 | \( y = \frac{6 - 2}{2} = 2 \) |

**Coordinates:**
- (0, 3)
- (2, 2)

### **Equation 2: \( 2x - 3y = 12 \)**

#### Step 1: Solve for \( y \)
Similarly, solve the second equation for \( y \):
\[
-3y = 12 - 2x
\]
\[
y = \frac{2x - 12}{3}
\]

#### Step 2: Create the T-Chart
Choose two values for \( x \) and find the corresponding \( y \) values.

| \( x \) | \( y \) |
|:---:|:---:|
| 0 | \( y = \frac{2(0) - 12}{3} = -4 \) |
| 3 | \( y = \frac{2(3) - 12}{3} = -2 \) |

**Coordinates:**
- (0, -4)
- (3, -2)

### **Step 3: Graph the Equations**
1. **For \( x + 2y = 6 \):**
   - Plot the points (0, 3) and (2, 2).
   - Draw a straight line through these points.

2. **For \( 2x - 3y = 12 \):**
   - Plot the points (0, -4) and (3, -2).
   - Draw a straight line through these points.

### **Step 4: Find the Intersection**
- The intersection point of the two lines is the solution to the system of equations.

**Solution:** The lines intersect at the point \((6, 0)\).

Would you like to see the graph or need further details?

### Related Questions:
1. How do you convert equations to slope-intercept form?
2. What are other methods to solve a system of equations besides graphing?
3. How do you verify if the intersection point is a solution to both equations?
4. What happens if you get a different pair of points to plot the same line?
5. How does changing the coefficients affect the graph of the equation?

### Tip:
Always check the intersection point by substituting it back into both original equations to confirm it's a valid solution.

Learn how to solve a system of linear equations by graphing. Explore the equations x + 2y = 6 and 2x - 3y = 12 with step-by-step T-chart creation, graph plotting, and intersection point determination. Suitable for Grades 7-9.

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