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

We are given the system: \[ \begin{aligned} x - 2y + 4 &= 0 \quad \text{(1)} \\ 2x - y - 4 &= 0 \quad \text{(2)} \end{aligned} \] To solve graphically, we express each equation in slope-intercept form: From (1): \[ x + 4 = 2y \Rightarrow y = \frac{1}{2}x + 2 \] From (2): \[ 2x - 4 = y \Rightarrow y = 2x - 4 \] Now we plot both lines on the coordinate plane. The point of intersection of the lines gives the solution. If they intersect at a single point, the system is consistent and has a unique solution. Solving algebraically to verify: \[ \begin{aligned} x - 2y + 4 &= 0 \quad \text{(i)} \\ 2x - y - 4 &= 0 \quad \text{(ii)} \end{aligned} \] Multiply (i) by 2: \[ 2x - 4y + 8 = 0 \] Subtract (ii): \[ (2x - 4y + 8) - (2x - y - 4) = 0 \\ -3y + 12 = 0 \Rightarrow y = 4 \] Substitute in (i): \[ x - 2(4) + 4 = 0 \Rightarrow x = 4 \] Hence, the system is consistent and has a unique solution: \((x, y) = (4, 4)\)