Question

Check whether the following system of equations is consistent or not. If consistent, solve graphically: \[ x - 2y + 4 = 0, \quad 2x - y - 4 = 0 \]

Hint:

A system of equations is consistent if the lines intersect at least once. Use slope-intercept form to graph easily.

Answer 1

The following system of equations is given: \[\begin{aligned}x - 2y + 4 &= 0 \quad \text{(1)} \\2x - y - 4 &= 0 \quad \text{(2)}\end{aligned}\] To solve graphically, we rewrite each equation in slope-intercept form. From equation (1): \[x + 4 = 2y \Rightarrow y = \frac{1}{2}x + 2\] From equation (2): \[2x - 4 = y \Rightarrow y = 2x - 4\] Plotting these lines reveals the solution as the point of intersection. A single intersection point indicates a consistent system with a unique solution. Algebraic verification: \[\begin{aligned}x - 2y + 4 &= 0 \quad \text{(i)} \\2x - y - 4 &= 0 \quad \text{(ii)}\end{aligned}\] Multiply (i) by 2: \[2x - 4y + 8 = 0\] Subtract (ii): \[(2x - 4y + 8) - (2x - y - 4) = 0 \\-3y + 12 = 0 \Rightarrow y = 4\] Substitute into (i): \[x - 2(4) + 4 = 0 \Rightarrow x = 4\] Therefore, the system is consistent and has the unique solution: \((x, y) = (4, 4)\)