Question

How do I solve a system of equations using an augmented matrix?

Answer 1

Hint: In this question, we have to find a method to solve a system of linear equations using the augmented matrices. An augmented matrix is the matrix of numbers in which each row represents a constant from one equation and each column represents a variable. First, we will change the system of equations in the form of a matrix and then change the matrix into the augmented matrix, which gives the solution of the system of equations, that is the required solution to the problem.

Complete step by step answer:
According to the question, we have to solve a system of equations using the augmented matrix.
As we know, an augmented matrix represents a constant in each row and a variable in each column. Therefore, to solve a system of equations, we will take an example to understand it properly.
Let us say the system of equations are $ 2x+4y=5 $ --- (1) and $ 10x+2y=5 $ ---- (2)
Now, we first change the system of equations in the form of the matrix, that is
 $ \left[ \begin{align}
  & \begin{matrix}
   02 & 4 & 5 \\
\end{matrix} \\
 & \begin{matrix}
   10 & 2 & 5 \\
\end{matrix} \\
\end{align} \right] $
Now, we will apply the elementary method in the above matrix, to solve it further. That is, we will make the first element of Row 2 equals to 0, by using the formula $ {{R}_{2}}={{R}_{2}}-5({{R}_{1}}) $ , we get
 $ \left[ \begin{align}
  & \begin{matrix}
   2 & +04 & +05 \\
\end{matrix} \\
 & \begin{matrix}
   0 & -18 & -20 \\
\end{matrix} \\
\end{align} \right] $
Now, we will make the first element of column 2 equals to 0, that is we will apply the operation $ {{R}_{1}}={{R}_{1}}+\dfrac{4}{18}{{R}_{2}} $ , therefore we get
 $ \left[ \begin{align}
  & \begin{matrix}
   2 & +00 & 0.55 \\
\end{matrix} \\
 & \begin{matrix}
   0 & -18 & -20 \\
\end{matrix} \\
\end{align} \right] $
Therefore, from the above matrix, we get the value of x and y, which is
 $ 2x=0.55 $ ------- (3) and $ -18y=-20 $ ------- (4)
Now, we will solve equation (3) by dividing 2 on both sides of the equation, we get
 $ \dfrac{2}{2}x=\dfrac{0.55}{2} $
 $ \Rightarrow x=0.275 $
Now, we will solve equation (4) by dividing -18 on both sides of the equation, we get
 $ \dfrac{-18}{-18}y=\dfrac{-20}{-18} $
 $ \Rightarrow y=1.12 $
Therefore, we get the value of variables using the augmented matrix in the system of equations.

Note:
While solving this problem, keep in mind the definition of augmented matrix and solve them step-by-step. After this step $ \left[ \begin{align}
  & \begin{matrix}
   2 & +04 & +05 \\
\end{matrix} \\
 & \begin{matrix}
   0 & -18 & -20 \\
\end{matrix} \\
\end{align} \right] $ , you can solve by forming them in terms of equations and then put the value of y in the first equation, to get the required result to the problem.