Question

Solve the following equations for x and y:
$152x\text{ }-\text{ 378}y\text{ = }-74,\text{ }-378x\text{ + 152}y\text{ = }-604.$
A. $x\text{ = }-2\text{ and }y\text{ = }-1$
B. $x\text{ = 2 and }y\text{ = 41}$
C. $x\text{ = 2 and }y\text{ = 1}$
D. $x\text{ = }-2\text{ and }y\text{ = 1}$

Answer 1

Hint: Solve the problem using substitution method. Find the value of one of the variables in the form of another variable using the first equation. Then put its value in the second equation and solve for that variable. Then putting the value of this variable, find the value of the other variable.

Complete step-by-step answer:
Let,
$\begin{align}
  & 152x\text{ }-\text{ 378}y\text{ = }-74\text{ }.....\text{(i)} \\
 & -378x\text{ + 152}y\text{ = }-604\text{ }.....\text{(ii)} \\
\end{align}$
In the equation (i),
$\begin{align}
  & 152x\text{ }-\text{ 378}y\text{ = }-74 \\
 & \Rightarrow \text{ 152}x\text{ + 74 = 378}y \\
 & \therefore \text{ }y\text{ = }\frac{152x\text{ + 74}}{378} \\
 & \text{ = }\frac{76x\text{ + 37}}{189}\text{ }....\text{(iii)} \\
\end{align}$
Putting the value of the variable y in the form of variable x in the equation (ii), we get,
\[\begin{align}
  & -378x\text{ + 152}\cdot \frac{\left( 76x\text{ + 37} \right)}{189}\text{ = }-604 \\
 & \Rightarrow \text{ }\left( -378x\cdot \text{189 + 76}x\cdot \text{152} \right)\text{ = }-604\cdot 189\text{ }-152\cdot 37 \\
 & \Rightarrow \text{ }-59890x\text{ = }-119780 \\
 & \therefore \text{ }x\text{ = }\frac{119780}{59890}\text{ = 2} \\
\end{align}\]
Now, putting the value of x = 2 in the equation (iii), we get the value of y.
For x = 2, we get,
$\begin{align}
  & y\text{ = }\frac{76\cdot 2\text{ + 37}}{189} \\
 & \text{ = }\frac{189}{189}\text{ = 1} \\
\end{align}$
Thus, solving the two equations by the method of substitution, we get, x = 2 and y = 1.
Hence, the correct answer is option C.

Note: The simultaneous equations can be solved by other alternative methods as well like the method of elimination, cross-multiplication etc. In the method of elimination taking the L.C.M. of the coefficients of the two variables, we need to express them in the lowest common multiple. Thereby, by transforming the coefficients of any of the two variables into the same value, we can eliminate that variable. Thus, we get the value of the other variable, putting which in any of the two equations, we also find the value of the first variable.