Write the point-slope form of the line's equation satisfying the given conditions. Then use the point-slope form of the equation to write the slope-intercept form of the equation Slope \( =5 \), passing through \( (7,4) \)

Certainly! Let's work through the problem step by step. ### **Given:** - **Slope (m):** 5 - **Point (x₁, y₁):** (7, 4) ### **1. Point-Slope Form** The **point-slope form** of a linear equation is useful when you know the slope of the line and a point that lies on the line. The formula is: \[ y - y_1 = m(x - x_1) \] **Plugging in the given values:** - \( m = 5 \) - \( x_1 = 7 \) - \( y_1 = 4 \) \[ y - 4 = 5(x - 7) \] This is the **point-slope form** of the equation for the line. ### **2. Converting to Slope-Intercept Form** The **slope-intercept form** is another way to express a linear equation, and it is given by: \[ y = mx + b \] where: - \( m \) is the slope - \( b \) is the y-intercept Let's convert the point-slope form we found into slope-intercept form. **Starting with the point-slope form:** \[ y - 4 = 5(x - 7) \] **Step-by-Step Conversion:** 1. **Distribute the slope (5) on the right-hand side:** \[ y - 4 = 5x - 35 \] 2. **Add 4 to both sides to solve for \( y \):** \[ y = 5x - 35 + 4 \] 3. **Combine like terms:** \[ y = 5x - 31 \] Now, the equation is in **slope-intercept form**: \[ y = 5x - 31 \] ### **Summary:** - **Point-Slope Form:** \( y - 4 = 5(x - 7) \) - **Slope-Intercept Form:** \( y = 5x - 31 \)