Solve System of Equations Graphically: $ y = x - 5 $ and $ y = -2x - 2 $

Question

Solve the following system of equations graphically: $ y = x - 5 $ and $ y = -2x - 2 $.

Accepted Answer

The solution is $ (1, -4) $ (or $ x = 1 $, $ y = -4 $)

Answer

1. **Graph $ y = x - 5 $:**\n   - Identify the y-intercept: $ b = -5 $, so the point is $ (0, -5) $.\n   - Use the slope $ m = 1 $ (rise 1, run 1) to find a second point: from $ (0, -5) $, move right 1 and up 1 to $ (1, -4) $, or right 5 and up 5 to $ (5, 0) $.\n   - Plot $ (0, -5) $ and $ (5, 0) $, then draw the line through them.\n 2. **Graph $ y = -2x - 2 $:**\n   - Identify the y-intercept: $ b = -2 $, so the point is $ (0, -2) $.\n   - Use the slope $ m = -2 $ (rise -2, run 1) to find a second point: from $ (0, -2) $, move right 1 and down 2 to $ (1, -4) $, or left 1 and up 2 to $ (-1, 0) $.\n   - Plot $ (0, -2) $ and $ (-1, 0) $, then draw the line through them.\n 3. **Find the Intersection:**\n   - The two lines intersect where their $ x $ and $ y $ values are equal. By plotting the lines, they cross at $ (1, -4) $.\n   - Verify: Substitute $ x = 1 $ into both equations:\n     - $ y = 1 - 5 = -4 $ (for $ y = x - 5 $)\n     - $ y = -2(1) - 2 = -4 $ (for $ y = -2x - 2 $)\n     - Both give $ y = -4 $, so $ (1, -4) $ is the solution.