Question

Question 4 (i)
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

The given equations are x + y = 5 and 2x + 2y = 10

They can be written as x + y - 5 = 0 and 2x + 2y - 10 = 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:

$a_1=1,b_1=1,and{c}_1=-5$
$a_2=2,b_2=2and{c}_2=-10$

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

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

$\frac{c_1}{c_2}=\frac{5}{10}=\frac{1}{2}$

Hence, $\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$

Therefore, these linear equations represent a coincident pair of lines and thus have infinite number of possible solutions.
Hence, the pair of linear equations is consistent.
x + y = 5
$\Rightarrow$ x = 5 - y
$\begin{array}{cccc}x& 4& 3& 2\\ y& 1& 2& 3\end{array}$
And, 2x + 2y = 10
$\Rightarrow$ $x=\frac{10-2y}{2}$
$\Rightarrow$ x = 5 - y
$\begin{array}{cccc}x& 4& 3& 2\\ y& 1& 2& 3\end{array}$
Graphical representation

From the figure, it can be observed that these lines are overlapping each other. Therefore, infinite solutions are possible for the given pair of equations.