Question

Which of the following pairs of linear equations are consistent/inconsistent?

If consistent, obtain the solution graphically:

$x+y=5,2x+2y=10$.

Answer 1

Step 1. check pairs of linear equations are consistent/inconsistent.

We have

$x+y=5,2x+2y=10$

Now,

$$\begin{gathered} \frac{a_1}{a_2}=\frac{1}{2} \\ \frac{b_1}{b_2}=\frac{1}{2} \\ \frac{c_1}{c_2}=\frac{1}{2} \\ \therefore \frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2} \end{gathered}$$

Thus, the equations are coincident and they have infinite number of possible solutions.

Step 2. Graphical solution.

For,$x+y=5$

$x$	$0$	$3$	$5$
$y$	$5$	$2$	$0$

For,$2x+2y=10$

$x$	$0$	$3$	$5$
$y$	$5$	$2$	$0$

The graphical representation of the equations is:

Here the lines are overlapping each other. Therefore, the equations have infinite possible solutions.

Hence, the given equations are consistent and have infinitely many solutions.