slope of \( \frac{1}{5} \) and through \( (-5,8) \)

To find the equation of a line with a slope of \( \frac{1}{5} \) that passes through the point \( (-5, 8) \), you can use the **point-slope form** of a linear equation. Here's a step-by-step guide: ### 1. Point-Slope Form The point-slope form of a line is given by: \[ y - y_1 = m(x - x_1) \] Where: - \( m \) is the slope of the line. - \( (x_1, y_1) \) is a point on the line. ### 2. Plug in the Given Values Given: - Slope \( m = \frac{1}{5} \) - Point \( (x_1, y_1) = (-5, 8) \) Substitute these values into the point-slope form: \[ y - 8 = \frac{1}{5}(x - (-5)) \] \[ y - 8 = \frac{1}{5}(x + 5) \] ### 3. Simplify to Slope-Intercept Form To express the equation in the slope-intercept form \( y = mx + b \), where \( b \) is the y-intercept, follow these steps: \[ y - 8 = \frac{1}{5}x + \frac{1}{5} \times 5 \] \[ y - 8 = \frac{1}{5}x + 1 \] \[ y = \frac{1}{5}x + 1 + 8 \] \[ y = \frac{1}{5}x + 9 \] ### 4. Final Equation The equation of the line in slope-intercept form is: \[ \boxed{y = \frac{1}{5}x + 9} \] ### 5. Alternative Forms - **Standard Form:** \( Ax + By = C \) To convert to standard form: \[ y = \frac{1}{5}x + 9 \] Multiply all terms by 5 to eliminate the fraction: \[ 5y = x + 45 \] Rearrange: \[ x - 5y + 45 = 0 \quad \text{or} \quad x - 5y = -45 \] Typically, standard form is written with positive \( A \): \[ \boxed{x - 5y = -45} \] - **Point-Slope Form:** As derived earlier: \[ \boxed{y - 8 = \frac{1}{5}(x + 5)} \] ### 6. Graphical Representation If you plot the line \( y = \frac{1}{5}x + 9 \): - **Slope \( \frac{1}{5} \):** For every 5 units you move to the right (positive direction on the x-axis), the line rises by 1 unit. - **Y-Intercept \( 9 \):** The line crosses the y-axis at the point \( (0, 9) \). ### 7. Verification Ensure that the point \( (-5, 8) \) lies on the line: \[ y = \frac{1}{5}x + 9 \] \[ 8 = \frac{1}{5}(-5) + 9 \] \[ 8 = -1 + 9 \] \[ 8 = 8 \quad \checkmark \] The calculation confirms that the point \( (-5, 8) \) satisfies the equation. --- Feel free to ask if you need further assistance or explanations!