Courses
Courses for Kids
Free study material
Offline Centres
More
Store

# If $19x - 17y = 55$ and $17x - 19y = 53$, then the value of $x - y$ is:A. $\dfrac{1}{3}$ B. $- 3$C. 3D. 5

Last updated date: 11th Aug 2024
Total views: 388.8k
Views today: 11.88k
Answer
Verified
388.8k+ views
Hint: Here, we are required to find the value of $x - y$. We would solve the given equations by any of the accepted methods and find the values of $x$ and $y$ respectively. Substituting their values in$x - y$, we would get our required answer.

Complete step-by-step answer:
The given two equations are:
$19x - 17y = 55$ ………..$\left( 1 \right)$
$17x - 19y = 53$………..$\left( 2 \right)$
We will multiply equation $\left( 1 \right)$ by 17, we get
$\left( {19x - 17y} \right) \times 17 = 55 \times 17$
$\Rightarrow \left( {19 \times 17} \right)x - {17^2}y = 55 \times 17$……………………….$\left( 3 \right)$
We will multiply equation $\left( 2 \right)$ by 19, we get
$\left( {17x - 19y} \right) \times 19 = 53 \times 19$
$\Rightarrow \left( {17 \times 19} \right)x \times {19^2}y = 53 \times 19$………………………$\left( 4 \right)$
Hence, subtracting the equations, we get,
Hence, subtracting the equation $\left( 4 \right)$ from equation $\left( 3 \right)$, we get,
$\left[ {\left( {19 \times 17} \right)x - {{17}^2}y} \right] - \left[ {\left( {19 \times 17} \right)x - {{19}^2}y} \right] = 55 \times 17 - 53 \times 19$
$\Rightarrow \left( {19 \times 17} \right)x - {17^2}y - \left( {19 \times 17} \right)x + {19^2}y = 55 \times 17 - 53 \times 19$
Solving the terms with same variables, we get,
$\Rightarrow \left( {{{19}^2} - {{17}^2}} \right)y = \left( {55 \times 17} \right) - \left( {53 \times 19} \right)$
Now, applying the identity $\left( {{a^2} - {b^2}} \right) = \left( {a + b} \right)\left( {a - b} \right)$
$\Rightarrow \left( {19 + 17} \right)\left( {19 - 17} \right)y = \left( {55 \times 17} \right) - \left( {53 \times 19} \right)$
Now, we would transport everything which is multiplying to the variable $y$ in the LHS to the denominator of RHS.
$\Rightarrow y = \dfrac{{\left( {55 \times 17} \right) - \left( {53 \times 19} \right)}}{{\left( {19 + 17} \right)\left( {19 - 17} \right)}}$
Now, we would solve this further,
$\Rightarrow y = \dfrac{{935 - 1007}}{{36 \times 2}}$
$\Rightarrow y = \dfrac{{ - 72}}{{72}}$
$\Rightarrow y = - 1$
Now, substituting this value in the equation,$19x - 17y = 55$, we get,
$19x - 17\left( { - 1} \right) = 55$
$19x + 17 = 55$
Subtracting 17 from both sides,
$19x = 38$
Dividing both sides by 19
$\Rightarrow x = 2$
Therefore, the required value of $x - y$ is:
$x - y = 2 - \left( { - 1} \right)$
$\Rightarrow x - y = 2 + 1 = 3$
Therefore, if $19x - 17y = 55$ and $17x - 19y = 53$, then the value of $x - y = 3$

Hence, option C is the required answer.

Note:
We can notice in the given two equations that the coefficient of $x$ in the first equation is the coefficient of $y$ in the second equation. Similarly, the coefficient of $y$ in the first equation is the coefficient of $x$ in the second equation.
Hence, there is another way to solve this question.
The given two equations are:
$19x - 17y = 55$ ………..$\left( 1 \right)$
$17x - 19y = 53$………..$\left( 2 \right)$
Now, adding the equations $\left( 1 \right)$ and $\left( 2 \right)$, we get,
$19x - 17y + 17x - 19y = 55 + 53$
$\Rightarrow 36x - 36y = 108$
Now, taking 36 common in the LHS,
$\Rightarrow 36\left( {x - y} \right) = 108$
Dividing both sides by 36
$\Rightarrow x - y = 3$…………………………$\left( 5 \right)$
Similarly, subtracting the equations $\left( 1 \right)$ and $\left( 2 \right)$, we get,
$19x - 17y - 17x + 19y = 55 - 53$
$\Rightarrow 2x + 2y = 2$
Now, taking 2 common in the LHS,
$\Rightarrow 2\left( {x + y} \right) = 2$
Dividing both sides by 2
$\Rightarrow x + y = 1$…………………………$\left( 6 \right)$
Now, adding the equations $\left( 5 \right)$ and $\left( 6 \right)$ using elimination method,
$\Rightarrow x - y + x + y = 3 + 1$
$\Rightarrow 2x = 4$
Dividing both sides by 2, we get
$\Rightarrow x = 2$
Substituting this value in equation $\left( 5 \right)$
$\Rightarrow x - y = 3$
$\Rightarrow 2 - y = 3$
$\Rightarrow y = 2 - 3 = - 1$
Therefore, the required value of $x - y$ is:
$x - y = 2 - \left( { - 1} \right)$
$\Rightarrow x - y = 2 + 1 = 3$
Hence, we could attempt this question by either of the two ways.