Question
Is the pair of linear equation consistent/inconsistent? If consistent, obtain the solution graphically: $x + y = 5, 2x + 2y = 10$
✓
Answer
$x + y = 5 ...(1)$
$2x + 2y = 10 ...(2)$
$\text { Here, } a_1=1, b_1=1, c_1=-5$
$ a_2=2, b_2=2, c_2=-10$
We see that $\frac{{{a_1}}}{{{a_2}}} = \frac{{{b_1}}}{{{b_2}}} = \frac{{{c_1}}}{{{c_2}}}$
Hence, the lines represented by the equations $(1)$ and $(2)$ are coincident.
Therefore, equations $(1)$ and $(2)$ have infinitely many common solutions, i.e., the given pair of linear equations is consistent.
Graphical Representation, we draw the graphs of the equations $(1)$ and $(2)$ by finding two solutions for each if the equations. These two solutions of the equations $(1)$ and $(2)$ are given below in table $1$ and table $2$ respectively.
For equation $ (1) x + y = 5 \Rightarrow y = 5 - x$
Table $1$ of solutions
For equations $(2) x + 2y = 10$
$\Rightarrow 2y = 10 - 2x$
$\Rightarrow y = \frac{{10 - 2x}}{2} \Rightarrow y = 5 - x$
Table $2$ of solutions
We plot the points $A(0, 5)$ and $B(5, 0)$ on a graph paper and join these points to form the line $AB$ representing the equation $(1)$ as shown in the figure, Also, we plot the points $C(1, 4)$ and $D (2, 3)$ on the same graph paper and join these points to form the line $CD$ representing the equation $(2)$ as shown in the same figure.
In the figure we observe that the two lines $AB$ and $CD$ coincide.
$2x + 2y = 10 ...(2)$
$\text { Here, } a_1=1, b_1=1, c_1=-5$
$ a_2=2, b_2=2, c_2=-10$
We see that $\frac{{{a_1}}}{{{a_2}}} = \frac{{{b_1}}}{{{b_2}}} = \frac{{{c_1}}}{{{c_2}}}$
Hence, the lines represented by the equations $(1)$ and $(2)$ are coincident.
Therefore, equations $(1)$ and $(2)$ have infinitely many common solutions, i.e., the given pair of linear equations is consistent.
Graphical Representation, we draw the graphs of the equations $(1)$ and $(2)$ by finding two solutions for each if the equations. These two solutions of the equations $(1)$ and $(2)$ are given below in table $1$ and table $2$ respectively.
For equation $ (1) x + y = 5 \Rightarrow y = 5 - x$
Table $1$ of solutions
| $x$ | $0$ | $5$ |
| $y$ | $5$ | $0$ |
$\Rightarrow 2y = 10 - 2x$
$\Rightarrow y = \frac{{10 - 2x}}{2} \Rightarrow y = 5 - x$
Table $2$ of solutions
| $x$ | $1$ | $2$ |
| $y$ | $4$ | $3$ |
In the figure we observe that the two lines $AB$ and $CD$ coincide.
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