Q 211 - Intermediate Algebra

Question

Solve Systems of Equations Using Matrices

In the following exercises, solve each system of equations using a matrix.

$\left\{\begin{array}{l}3x+y=2\\ x-y=2\end{array}\right.$

Step-by-Step Solution

Answer

The solution of the system is $(1,-1)$

Step 1. Given

A system of linear equations is given as-

$\left\{\begin{array}{l}3x+y=2\\ x-y=2\end{array}\right.$

Step 2. Concept Used

First, we will convert the given system of linear equations into its argument matrix. After that we will apply row transformation accordingly. 

And we will to make zeroes in the matrix at least one through which we can find the value of one of the variable. And then we will find the value of other variable by using value of first one.

Step 3. Calculation

The solution of the system is-

$\left\{\begin{array}{l}3x+y=2\\ x-y=2\end{array}\right.$

Accordingly, the argument matrix is-

$\left[\begin{array}{rrr}3& 1& 2\\ 1& -1& 2\end{array}\right]$

Now add row $1$ and row $2$,

$\left[\begin{array}{rrr}3& 1& 2\\ 1& -1& 2\end{array}\right]\stackrel{R_1+R_2}{⟶}\left[\begin{array}{rrr}1& -1& 2\\ 3& 1& 2\end{array}\right]$

$$\begin{gathered} \left[\begin{array}{rrr}3+1& 1+(-1)& 2+2\\ 1& -1& 2\end{array}\right] \\ \to \left[\begin{array}{rrr}4& 0& 4\\ 1& -1& 2\end{array}\right] \end{gathered}$$

Step 4. Further Calculation

Now the corresponds equations are-

$$\begin{gathered} 4x=4 \\ x-y=2 \end{gathered}$$

Use substitution method for remaining value,

$$\begin{gathered} 4x=4\to x=1 \\ x-y=2 \\ 1-y=2 \\ y=1-2 \\ y=-1 \end{gathered}$$

Step 5. Verification

Verify the equation,

$$\begin{gathered} 3 x+y=2 \\ 3\left(1\right)-1=2 \\ 3-1=2 \\ 2=2 \end{gathered}$$

LHS is equal to RHS