4. \( (0,-8) \) and \( (3,2) \)

The problem provides two points: \( (0,-8) \) and \( (3,2) \). These points can be used to find the equation of a line passing through them. To find the equation of a line passing through two points \( (x_1, y_1) \) and \( (x_2, y_2) \), we can use the point-slope form of the equation of a line: \[ y - y_1 = m(x - x_1) \] where \( m \) is the slope of the line. First, let's calculate the slope of the line passing through the points \( (0,-8) \) and \( (3,2) \): \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Substitute the given points into the formula to find the slope. Calculate the value by following steps: - step0: Calculate: \(\frac{\left(2-\left(-8\right)\right)}{\left(3-0\right)}\) - step1: Remove the parentheses: \(\frac{2-\left(-8\right)}{3-0}\) - step2: Remove the parentheses: \(\frac{2+8}{3-0}\) - step3: Add the numbers: \(\frac{10}{3-0}\) - step4: Remove 0: \(\frac{10}{3}\) The slope of the line passing through the points \( (0,-8) \) and \( (3,2) \) is \( \frac{10}{3} \) or approximately 3.33. Now that we have the slope, we can choose one of the points to substitute into the point-slope form of the equation of a line to find the equation of the line passing through the two points. Let's choose the point \( (0,-8) \) and substitute the values into the equation: \[ y - (-8) = \frac{10}{3}(x - 0) \] Simplify the equation to get the final equation of the line passing through the points \( (0,-8) \) and \( (3,2) \). To find the equation of the line in slope-intercept form, we need to rearrange the equation \( y + 8 = \frac{10}{3}x \) to isolate \( y \). \[ y = \frac{10}{3}x - 8 \] Therefore, the equation of the line passing through the points \( (0,-8) \) and \( (3,2) \) is \( y = \frac{10}{3}x - 8 \).