1 Determine the linear equation for the line that goes through the following coordinates: \( \begin{array}{l}\text { Q.9.1.1 } \\ \text { Q.5.10) and }(-3,4) \\ (-4,-22) \text { and }(-6,-34)\end{array} \)

To determine the linear equation for the line that goes through the given coordinates, we will follow these steps: 1. **Identify the coordinates**: The points given are: - \( A(-3, 4) \) - \( B(-4, -22) \) - \( C(-6, -34) \) 2. **Calculate the slope (m)**: The slope of a line through two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] 3. **Use the point-slope form of the equation**: Once we have the slope, we can use the point-slope form of the equation: \[ y - y_1 = m(x - x_1) \] 4. **Convert to slope-intercept form (if needed)**: The slope-intercept form is given by: \[ y = mx + b \] Now, let's calculate the slope using the first two points \( A(-3, 4) \) and \( B(-4, -22) \). ### Step 1: Calculate the slope between points A and B Using the coordinates of points A and B: - \( x_1 = -3, y_1 = 4 \) - \( x_2 = -4, y_2 = -22 \) Calculating the slope \( m \): \[ m = \frac{-22 - 4}{-4 - (-3)} = \frac{-26}{-1} = 26 \] ### Step 2: Use the point-slope form Using point A(-3, 4) and the slope \( m = 26 \): \[ y - 4 = 26(x + 3) \] ### Step 3: Convert to slope-intercept form Expanding the equation: \[ y - 4 = 26x + 78 \] \[ y = 26x + 82 \] Now, let's check if the other points \( B(-4, -22) \) and \( C(-6, -34) \) also lie on this line. ### Step 4: Verify with point B Substituting \( x = -4 \) into the equation: \[ y = 26(-4) + 82 = -104 + 82 = -22 \] This matches point B. ### Step 5: Verify with point C Substituting \( x = -6 \) into the equation: \[ y = 26(-6) + 82 = -156 + 82 = -74 \] This does not match point C, indicating that points B and C do not lie on the same line as A and B. ### Conclusion The linear equation for the line that goes through points A and B is: \[ y = 26x + 82 \] However, points B and C do not lie on the same line as A and B. If you need the equation for the line through points B and C, please let me know!