Q18. - Algebra 1

Question

Use an augmented matrix to solve each system of equations.

$\begin{array}{c}\mathbf{y}\mathbf{=}\mathbf{3}\mathbf{x}\\ \mathbf{x}\mathbf{+}\mathbf{2}\mathbf{y}\mathbf{=}\mathbf{21}\end{array}$

Step-by-Step Solution

Answer

The solution of the given system of equations is $\mathbf{(}\mathbf{3}\mathbf{,}\mathbf{9}\mathbf{)}$.

Step 1. Write the augmented matrix for the given system of equations.

The given system of equations is:

$\begin{array}{c}y=3x\\ x+2y=21\end{array}$

The given system of equations can be written as:

$\begin{array}{c}3x-y=0\\ x+2y=21\end{array}$

Place the coefficients of the equations and the constant terms into the matrix to form the augmented matrix for the given system of equations.

$\left[\begin{array}{ccc}3& -1& 0\\ 1& 2& 21\end{array}\right]$

Step 2. Solve the given system of equations.

The augmented matrix is:

$\left[\begin{array}{ccc}3& -1 & |0\\ 1& 2 & |21\end{array}\right]$

Interchange $R_2$ and $R_1$.

$\left[\begin{array}{ccc}1& 2& 21\\ 3& -1& 0\end{array}\right]$

Apply $R_2\to R_2-3R_1$,

$\left[\begin{array}{ccc}1& 2& 21\\ 0& -7& -63\end{array}\right]$

Apply ,

$\left[\begin{array}{ccc}1& 2& 21\\ 0& 1& 9\end{array}\right]$

Apply $R_1\to R_1-2R_2$,

$\left[\begin{array}{ccc}1& 0& 3\\ 0& 1& 9\end{array}\right]$

Therefore, the solution of the given system of equations is $(3,9)$.