To find the coordinates of the mid-point of the x and y intercepts of the line given by the equation \(2y = 4x - 8\), we will follow these steps: ### Step 1: Rewrite the Equation in Slope-Intercept Form First, we need to rewrite the equation in a more familiar form, which is the slope-intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. Starting with the original equation: \[ 2y = 4x - 8 \] We divide every term by 2 to isolate \(y\): \[ y = 2x - 4 \] ### Step 2: Find the Y-Intercept The y-intercept occurs where \(x = 0\). We can find it by substituting \(x = 0\) into the equation: \[ y = 2(0) - 4 = -4 \] Thus, the y-intercept is at the point \((0, -4)\). ### Step 3: Find the X-Intercept The x-intercept occurs where \(y = 0\). We can find it by substituting \(y = 0\) into the equation: \[ 0 = 2x - 4 \] Adding 4 to both sides gives: \[ 2x = 4 \] Dividing by 2 results in: \[ x = 2 \] Thus, the x-intercept is at the point \((2, 0)\). ### Step 4: Find the Mid-Point of the Intercepts Now that we have both intercepts: - Y-intercept: \((0, -4)\) - X-intercept: \((2, 0)\) We can find the mid-point of these two points using the mid-point formula: \[ \text{Mid-point} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \] Substituting in our intercept coordinates: \[ \text{Mid-point} = \left( \frac{0 + 2}{2}, \frac{-4 + 0}{2} \right) = \left( \frac{2}{2}, \frac{-4}{2} \right) = (1, -2) \] ### Conclusion The coordinates of the mid-point of the x and y intercepts of the line \(2y = 4x - 8\) are \((1, -2)\). ### Explanation of Options - **Option A: (2, 0)** - This is the x-intercept, not the mid-point. - **Option B: (1, -2)** - This is the correct answer, as calculated above. - **Option C: (-1, -2)** - This point does not correspond to either intercept or the mid-point. - **Option D: (1, 2)** - This point does not correspond to either intercept or the mid-point. ### Revision Summary - The y-intercept is found by setting \(x = 0\) in the equation. - The x-intercept is found by setting \(y = 0\) in the equation. - The mid-point of two points is calculated using the mid-point formula. - The correct mid-point of the intercepts for the line \(2y = 4x - 8\) is \((1, -2)\).