Use the given conditions to write an equation for the line in point-slope form and slope-intercept form. Passing through \( (-5,-2) \) and \( (5,12) \) Type the point-slope form of the equation of the line. \( \square \) (Use integers or simplified fractions for any numbers in the equation.) Type the slope-intercept form of the equation of the line. \( \square \) Use integers or simplified fractions for any numbers in the equation.)

To find the equation of the line, we first calculate the slope using the two given points \((-5, -2)\) and \((5, 12)\). 1. Calculate the slope \( m \): \[ m = \frac{12 - (-2)}{5 - (-5)} = \frac{12 + 2}{5 + 5} = \frac{14}{10} = \frac{7}{5} \] 2. Write the equation in point-slope form using the point \((-5,-2)\): \[ y - (-2) = \frac{7}{5}(x - (-5)) \] Simplify the equation: \[ y + 2 = \frac{7}{5}(x + 5) \] 3. Convert the equation to slope-intercept form. Start with: \[ y + 2 = \frac{7}{5}(x + 5) \] Distribute the \(\frac{7}{5}\): \[ y + 2 = \frac{7}{5}x + \frac{7}{5}\cdot 5 \] Simplify: \[ \frac{7}{5}\cdot 5 = 7 \] Thus: \[ y + 2 = \frac{7}{5}x + 7 \] Subtract 2 from both sides: \[ y = \frac{7}{5}x + 7 - 2 = \frac{7}{5}x + 5 \] The equations are: - Point-slope form: \[ y + 2 = \frac{7}{5}(x + 5) \] - Slope-intercept form: \[ y = \frac{7}{5}x + 5 \]