Polnt A 31,19, Point in 29,132 Slopel Point Stope Form: Slope Intersept Form: Standard Form:

To find the slope and the equations of the line that passes through the two points \( A(31, 19) \) and \( B(29, 132) \), we can follow these steps: ### Step 1: Calculate the Slope (m) The formula for the slope \( m \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Substituting the coordinates of points A and B: - \( (x_1, y_1) = (31, 19) \) - \( (x_2, y_2) = (29, 132) \) \[ m = \frac{132 - 19}{29 - 31} = \frac{113}{-2} = -56.5 \] ### Step 2: Point-Slope Form The point-slope form of a line is given by: \[ y - y_1 = m(x - x_1) \] Using point A \( (31, 19) \) and the slope \( m = -56.5 \): \[ y - 19 = -56.5(x - 31) \] ### Step 3: Slope-Intercept Form To convert the point-slope form to slope-intercept form \( y = mx + b \): 1. Distribute the slope: \[ y - 19 = -56.5x + 1751.5 \] 2. Add 19 to both sides: \[ y = -56.5x + 1751.5 + 19 \] \[ y = -56.5x + 1770.5 \] ### Step 4: Standard Form The standard form of a line is \( Ax + By = C \). We can rearrange the slope-intercept form to get it into standard form: Starting from: \[ y = -56.5x + 1770.5 \] Rearranging gives: \[ 56.5x + y = 1770.5 \] To eliminate the decimal, we can multiply through by 2: \[ 113x + 2y = 3541 \] ### Summary of Results 1. **Slope (m)**: \(-56.5\) 2. **Point-Slope Form**: \(y - 19 = -56.5(x - 31)\) 3. **Slope-Intercept Form**: \(y = -56.5x + 1770.5\) 4. **Standard Form**: \(113x + 2y = 3541\)