Question

Solve the following pairs of equations:
$\begin{align}
  & 7x-15y=12 \\
 & x+2y=3 \\
\end{align}$

Answer 1

Hint: We have a given set of the equation:
$\begin{align}
  & 7x-15y=12......(1) \\
 & x+2y=3......(2) \\
\end{align}$
We need to find the value of x and y. To get the values of x and y we need to solve the given set of linear equations. We can solve a set of equations by the substitution method. In the substitution method, we get the value of one variable in terms of other variables from one equation and substitute it in the other equation to get the value of another variable. Here, we will express y in terms of x by rearranging the second equation and then proceed.

Complete step-by-step solution:
We have a given set of equation:
$\begin{align}
  & 7x-15y=12.................(1) \\
 & x+2y=3.......................(2) \\
\end{align}$
So, by applying substitution method for the given set of linear equations:
From equation (2), we can write:
$\begin{align}
  & x+2y=3 \\
 & \Rightarrow 2y=3-x \\
 & \Rightarrow y=\dfrac{3-x}{2}...............(3) \\
\end{align}$
Now, we get the value of y in terms of x.
Put the value of y in equation (1), we get:
$\begin{align}
  & \Rightarrow 7x-15y=12 \\
 & \Rightarrow 7x-15\left( \dfrac{3-x}{2} \right)=12 \\
 & \Rightarrow 14x-15\left( 3-x \right)=24 \\
 & \Rightarrow 14x+15x=24+45 \\
 & \Rightarrow 29x=69 \\
 & \Rightarrow x=\dfrac{69}{29}....................(4) \\
\end{align}$
Put the value of x in equation (3) to get the value of y:
So, we have:
\[\begin{align}
  & y=\dfrac{3-\left( \dfrac{69}{29} \right)}{2} \\
 & =\dfrac{87-69}{58} \\
 & =\dfrac{18}{58} \\
 & =\dfrac{9}{29}.....(5)
\end{align}\]
Hence, $x=\dfrac{69}{29}$and $y=\dfrac{9}{29}$

Note: There is another method to solve a given set of linear equations, i.e. Elimination method. In the elimination method, we multiply or divide both the equations with a number to eliminate one variable in both the equations by adding or subtracting them and get the value of another value. Then we put the value in one of the equations and get the value of the eliminated variable.
For the given set of equations:
$\begin{align}
  & 7x-15y=12......(1) \\
 & x+2y=3......(2) \\
\end{align}$
Multiply equation (2) by 7, we get:
$\begin{align}
  & 7x-15y=12......(1) \\
 & 7x+14y=21......(3) \\
\end{align}$
Subtract equation (1) from (3); we get:
$\begin{align}
  & \Rightarrow 29y=9 \\
 & \Rightarrow y=\dfrac{9}{29} \\
\end{align}$
Now put the value of y in any of the above equation to get the value of x.
Let’s put value of y in equation (1), we get:
$\begin{align}
  & \Rightarrow x+2y=3 \\
 & \Rightarrow x+2\left( \dfrac{9}{29} \right)=3 \\
 & \Rightarrow 29x+18=87 \\
 & \Rightarrow x=\dfrac{69}{29} \\
\end{align}$