Question

If 7x+9y=42 and 9x+7y=22, then find the value of x+y.
a) 1
b) 2
c) 4
d) 3

Answer 1

Hint: Here, in this given question, we can use the elimination method of solving two linear equations with two variables to solve this given question. First of all, we must make the coefficients of one variable equal by taking the Lowest Common Multiple(LCM) and multiplying both the equations as required. Then we can subtract any one equation from the other and get the value of one variable. Then we may put in any of the equations to get the value of the other variable. Thereafter, in order to get our answer, we may add them.

Complete step-by-step solution -
In this question, let 7x+9y=42…………….(1.1) and
9x+7y=22…………(1.2).
Now, let us make the coefficients of x equal as 63 that is the LCM of 7 and 9.
$equation(1.1)\times 9=63x+81y=378..............(1.3)$
$equation(1.2)\times 7=63x+49y=154..............(1.4)$
Now, subtracting equation (1.4) from equation (1.3), we get,
\[\begin{align}
  & equation(1.3)-equation(1.4)=63x+81y-\left( 63x+49y \right)=378-154 \\
 & \Rightarrow 32y=224 \\
 & \Rightarrow y=\dfrac{224}{32} \\
 & \Rightarrow y=7.........(1.5) \\
\end{align}\]
Now, putting value of y from equation (1.5) in equation (1.2), we get,
\[\begin{align}
  & 9x+7y=22 \\
 & \Rightarrow 9x+7\times 7=22 \\
 & \Rightarrow 9x=22-49 \\
 & \Rightarrow x=\dfrac{-27}{9} \\
 & \Rightarrow x=-3.............(1.6) \\
\end{align}\]
Now, \[x+y=\left( -3 \right)+7=4.............(1.7)\]
So, from equation (1.7), we get the value of x+y as 4.
Therefore, the correct option to the given question is option (c) which is equal to 4.

Note: We may have also solved these two linear equations in two variables by using other methods such as by the method of substitution or graphically and must have arrived at the same answer as we have got here. If we were to use the method of substitution, we could have substituted the value of x in terms of y from the first equation in the second equation. Then, we would have got the value of y. Then re-substituting the obtained value of y, we could have got the value of x. After which we could have easily found out x+y.