The line's slope is \( -\frac{1}{4} \) and it passes through the point \( (10,-9) \)

To determine the equation of a line given its slope and a specific point it passes through, you can use the **point-slope form** and then convert it to other forms as needed. Here's a step-by-step guide based on the information provided: - **Slope (m):** \(-\frac{1}{4}\) - **Point (x₁, y₁):** \((10, -9)\) ### 1. Point-Slope Form The **point-slope form** of a line is given by: \[ y - y_1 = m(x - x_1) \] **Plugging in the values:** \[ y - (-9) = -\frac{1}{4}(x - 10) \] \[ y + 9 = -\frac{1}{4}x + \frac{10}{4} \] \[ y + 9 = -\frac{1}{4}x + 2.5 \] ### 2. Slope-Intercept Form To express the equation in **slope-intercept form** (\(y = mx + b\)), solve for \(y\): \[ y = -\frac{1}{4}x + 2.5 - 9 \] \[ y = -\frac{1}{4}x - 6.5 \] Alternatively, using fractions: \[ y = -\frac{1}{4}x - \frac{13}{2} \] ### 3. Standard Form The **standard form** of a line is \(Ax + By = C\), where \(A\), \(B\), and \(C\) are integers, and \(A\) is non-negative. Starting from the slope-intercept form: \[ y = -\frac{1}{4}x - 6.5 \] Multiply every term by 4 to eliminate the fraction: \[ 4y = -x - 26 \] Rearrange to standard form: \[ x + 4y = -26 \] ### Summary of the Equation in Different Forms 1. **Point-Slope Form:** \[ y + 9 = -\frac{1}{4}(x - 10) \] 2. **Slope-Intercept Form:** \[ y = -\frac{1}{4}x - 6.5 \] or \[ y = -\frac{1}{4}x - \frac{13}{2} \] 3. **Standard Form:** \[ x + 4y = -26 \] You can use any of these forms depending on the context or the specific requirements of your problem.