Solve , graphically :
` 15 x - 8y =29 `
` 17x +12y =75`

Answer

To solve the equations graphically, we will follow these steps:

1. **Write the equations**:
The equations given are:
\[ 15x - 8y = 29 \quad \text{(1)} \]
\[ 17x + 12y = 75 \quad \text{(2)} \]

2. **Find the x-intercept and y-intercept for the first equation**:
- To find the x-intercept, set \(y = 0\) in equation (1):
\[ 15x - 8(0) = 29 \implies 15x = 29 \implies x = \frac{29}{15} \approx 1.93 \]
- To find the y-intercept, set \(x = 0\) in equation (1):
\[ 15(0) - 8y = 29 \implies -8y = 29 \implies y = -\frac{29}{8} \approx -3.625 \]

3. **Plot the points for the first equation**:
- The x-intercept is approximately (1.93, 0).
- The y-intercept is approximately (0, -3.625).
- Draw the line through these points.

4. **Find the x-intercept and y-intercept for the second equation**:
- To find the x-intercept, set \(y = 0\) in equation (2):
\[ 17x + 12(0) = 75 \implies 17x = 75 \implies x = \frac{75}{17} \approx 4.41 \]
- To find the y-intercept, set \(x = 0\) in equation (2):
\[ 17(0) + 12y = 75 \implies 12y = 75 \implies y = \frac{75}{12} \approx 6.25 \]

5. **Plot the points for the second equation**:
- The x-intercept is approximately (4.41, 0).
- The y-intercept is approximately (0, 6.25).
- Draw the line through these points.

6. **Find the point of intersection**:
- To find the intersection point, we can solve the equations simultaneously. We can use the method of elimination.
- Multiply equation (1) by 12 and equation (2) by 8 to eliminate \(y\):
\[ 12(15x - 8y) = 12(29) \implies 180x - 96y = 348 \quad \text{(3)} \]
\[ 8(17x + 12y) = 8(75) \implies 136x + 96y = 600 \quad \text{(4)} \]

7. **Add equations (3) and (4)**:
\[ 180x - 96y + 136x + 96y = 348 + 600 \]
\[ 316x = 948 \implies x = \frac{948}{316} = 3 \]

8. **Substitute \(x = 3\) back into one of the original equations to find \(y\)**:
- Using equation (1):
\[ 15(3) - 8y = 29 \implies 45 - 8y = 29 \implies -8y = 29 - 45 \implies -8y = -16 \implies y = 2 \]

9. **Conclusion**:
The solution to the system of equations is:
\[ x = 3, \quad y = 2 \]