Question

Question 4 (iii)
Which of the following pairs of linear equations are consistent/ inconsistent? If consistent, obtain the solution graphically:
2x + y - 6 = 0, 4x - 2y - 4 = 0

Answer 1

The given equations are 2x + y - 6 = 0 and 4x - 2y - 4 = 0.

Comparing these equations with $a_{1}x+b_{1}y+c_1=0$ and $a_{2}x+b_{2}y+c_2=0$, we get:

$a_1=2,b_1=1,and{c}_1=-6$
$a_2=4,b_2=-2and{c}_2=-4$

$\frac{a_1}{a_2}=\frac{2}{4}$ = $\frac{1}{2}$

$\frac{b_1}{b_2}=\frac{1}{-2}$ = -$\frac{1}{2}$ and

$\frac{c_1}{c_2}=\frac{-6}{-4}=\frac{3}{2}$

Hence, $\frac{a_1}{a_2}\ne \frac{b_1}{b_2}$
Therefore, these linear equations represent intersecting lines and thus have only one possible solution.
Hence, the pair of linear equations is consistent.

Graphical representation:
2x + y - 6 = 0
y = 6 - 2x
$\begin{array}{cccc}x& 0& 1& 2\\ y& 6& 4& 2\end{array}$
And, 4x - 2y -4 = 0
$\Rightarrow$ $y=\frac{4x-4}{2}$
$\Rightarrow$ y = 2x - 2
$\begin{array}{cccc}x& 1& 2& 3\\ y& 0& 2& 4\end{array}$
Graphical representation

From the figure, it can be observed that these lines are intersecting each other at the only one point i.e., (2,2).

Thus, x = 2 and y = 2 is the solution for the given pair of equations.