Question

Check whether the following pair of linear equations has unique solution or not? If so, solve it by cross-multiplication method:

$23 x - 17 y + 11 = 0$ and $31 x + 13 y - 57 = 0$

Accepted Answer

Given equations are:

$23 x - 17 y + 11 = 0$

$31 x + 13 y - 57 = 0$

Comparing the above equations with the below general form, we get

$a_{1} x + b_{1} y + c_{1} = 0 a_{2} x + b_{2} y + c_{2} = 0$

$a_{1} = 23, b_{1} = - 17, c_{1} = 11$

$a_{2} = 31, b_{2} = 13, c_{2} = - 57$

$\Rightarrow \frac{a_{1}}{a_{2}} = \frac{23}{31}, \frac{b_{1}}{b_{2}} = \frac{- 17}{13}, \frac{c_{1}}{c_{2}} = \frac{- 11}{57}$

From the above comparison, we can clearly say that $\frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}}$

Hence, there will be unique solution for the given pair of linear equations.

Now, by using the cross-multiplication method, the unique solution is given by $\frac{x}{b_{1} c_{2} - b_{2} c_{1}} = \frac{y}{c_{1} a_{2} - c_{2} a_{1}} = \frac{1}{a_{1} b_{2} - a_{2} b_{1}}$ .

So, $\frac{x}{b_{1} c_{2} - b_{2} c_{1}} = \frac{1}{a_{1} b_{2} - a_{2} b_{1}}$

$\Rightarrow \frac{x}{(- 17) \times (- 57) - 13 \times (11)} = \frac{1}{23 \times 13 - 31 \times (- 17)}$

$\Rightarrow \frac{x}{969 - 143} = \frac{1}{299 + 527}$

$\Rightarrow \frac{x}{826} = \frac{1}{826}$

$\Rightarrow x = 1$ .

And, $\frac{y}{c_{1} a_{2} - c_{2} a_{1}} = \frac{1}{a_{1} b_{2} - a_{2} b_{1}}$

$\Rightarrow \frac{y}{11 \times 31 - 23 \times (- 57)} = \frac{1}{23 \times 11 - 31 \times (- 17)}$

$\Rightarrow \frac{y}{341 + 1311} = \frac{1}{299 + 527}$

$\Rightarrow \frac{y}{1652} = \frac{1}{826}$

$\Rightarrow y = \frac{1652}{826} = 2$

Therefore, the required unique solution is $x = 1$ and $y = 2$ .

M L Aggarwal Solutions for Chapter: Pair of Linear Equations in Two Variables, Exercise 9: Chapter Test

M L Aggarwal Mathematics Solutions for Exercise - M L Aggarwal Solutions for Chapter: Pair of Linear Equations in Two Variables, Exercise 9: Chapter Test

Questions from M L Aggarwal Solutions for Chapter: Pair of Linear Equations in Two Variables, Exercise 9: Chapter Test with Hints & Solutions

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