Question

How do you use the addition and subtraction method to solve a linear system?

Answer 1

Hint: We will use addition and subtraction methods to solve the linear equation. In this method we will eliminate one variable by addition or subtraction to find the value of another variable. Then by substituting the value and simplifying the obtained equation we will get the values of both the variables.

Complete step-by-step solution:
We have to explain the addition and subtraction method to solve a linear system.
We know that a linear equation in two variables is an equation which has two unknown variables. We have different methods to solve a linear system. In addition and subtraction method or also known as elimination method, one variable is eliminated by adding or subtracting both the equations. Then we will substitute the obtained value in one of the equations to get the value of another variable.
Let us understand this method by taking an example:
We have the following system of linear equations and we have to solve it by using the addition subtraction method.
$\begin{align}
  & x+y=2 \\
 & x-y=4 \\
\end{align}$
Now, let us add two equations we will get
\[\begin{align}
  & \Rightarrow \left( x+y \right)+\left( x-y \right)=2+4 \\
 & \Rightarrow 2x=6 \\
 & \Rightarrow x=\dfrac{6}{2} \\
 & \Rightarrow x=3 \\
\end{align}\]
Now, substituting the value of x in one of the equation we will get
$\begin{align}
  & \Rightarrow 3+y=2 \\
 & \Rightarrow y=2-3 \\
 & \Rightarrow y=-1 \\
\end{align}$
Hence we get the value of two variables.

Note: In this particular method to eliminate a variable the coefficient of the variable must be the same. To make coefficients of variables equal we will multiply or divide the equation by some constant. For example if we have following system of linear equations:
$\begin{align}
  & 3x+2y=2.........(i) \\
 & x-y=4............(ii) \\
\end{align}$
Before adding or subtracting the equations first we need to multiply the equation (ii) by 2 to make the coefficients of y equal. Then we will get
$\begin{align}
  & \Rightarrow 2x-2y=4\times 2 \\
 & \Rightarrow 2x-2y=8...........(iii) \\
\end{align}$
Now, adding the equation (i) and equation (iii) we will get
$\begin{align}
  & \Rightarrow \left( 3x+2y \right)+\left( 2x-2y \right)=2+8 \\
 & \Rightarrow 3x+2y+2x-2y=10 \\
 & \Rightarrow 5x=10 \\
 & \Rightarrow x=\dfrac{10}{5} \\
 & \Rightarrow x=2 \\
\end{align}$
Then we will solve the remaining part as explained in the solution.