Question

Solve for $x$ and $y$:
$\dfrac{x}{2} - \dfrac{y}{9} = 6$, $\dfrac{x}{7} + \dfrac{y}{3} = 5$

Answer 1

Hint: First we will take the LCM of both the linear equations given and convert them in such a way so that the denominator is $1$, hence it will be easier for us to do the calculations. Then we will multiply the required equation with a scalar quantity so that we can make the coefficient of one of the variables equal in both the equations, so that further on performing subtraction or addition (as per the requirement of the question) we can get one variable cancelled. Then we can easily find the value of the other variable, and then by substituting its value in any of the equations, the value of the other variable can also be achieved.

Complete step-by-step answer:
 Given, $\dfrac{x}{2} - \dfrac{y}{9} = 6 - - - \left( 1 \right)$
$\dfrac{x}{7} + \dfrac{y}{3} = 5 - - - \left( 2 \right)$
Now, simplifying the equations, we get,
From $\left( 1 \right)$,
$\dfrac{x}{2} - \dfrac{y}{9} = 6$
Taking, LCM on left hand side, we get,
$ \Rightarrow \dfrac{{9x - 2y}}{{18}} = 6$
Now, cross multiplying, we get,
\[ \Rightarrow 9x - 2y = 108 - - - \left( 3 \right)\]
Now, from $\left( 2 \right)$, we get,
$\dfrac{x}{7} + \dfrac{y}{3} = 5$
Taking, LCM on left hand side, we get,
$ \Rightarrow \dfrac{{3x + 7y}}{{21}} = 5$
Now, cross multiplying, we get,
$ \Rightarrow 3x + 7y = 105 - - - \left( 4 \right)$
Now, to solve the equations by the method of elimination, we will multiply $\left( 4 \right)$ by 3 and subtract from $\left( 3 \right)$.
$\therefore \left( 3 \right) - 3 \times \left( 4 \right)$, we get,
$\left( {9x - 2y} \right) - 3 \times \left( {3x + 7y} \right) = \left( {108} \right) - 3 \times \left( {105} \right)$
Now, simplifying, we get,
$ \Rightarrow \left( {9x - 2y} \right) - \left( {9x + 21y} \right) = \left( {108} \right) - \left( {315} \right)$
Now, opening the brackets, we get,
\[ \Rightarrow 9x - 2y - 9x - 21y = 108 - 315\]
\[ \Rightarrow - 23y = - 207\]
Now, dividing both sides by $ - 23$, we get,
\[ \Rightarrow y = \dfrac{{ - 207}}{{ - 23}}\]
\[ \Rightarrow y = 9\]
Now, substituting this value in $\left( 4 \right)$, we get,
$3x + 7\left( 9 \right) = 105$
Now, simplifying, we get,
$ \Rightarrow 3x + 63 = 105$
Subtracting $63$ from both sides of the equation, we get,
$ \Rightarrow 3x = 105 - 63$
$ \Rightarrow 3x = 42$
Now, dividing both sides by $3$, we get,
$ \Rightarrow x = \dfrac{{42}}{3} = 14$
Therefore, the values of $x$ and $y$ are $14$ and $9$ respectively.

Note: We can also solve this problem by other methods of solving linear equations viz. substitution method and cross multiplication method. But the substitution method can be very complicated on calculations and the cross multiplication method has a higher risk of mistakes, so using the elimination method is a better and easier method.