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

2x + y - 6 = 0, 4x - 2y - 4 = 0

Comparing these equations with,
$a_{1}x+b_{1}y+c_1=0$
$a_{2}x+b_{2}y+c_2=0$
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 are intersecting each other at one point and thus have only one possible solution.
Hence, the pair of linear equations is consistent.
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) which is the solution for the given pair of equations.