Question

In Problems 37–72, solve each system of equations using matrices (row operations). If the system has no solution, say that it is inconsistent.

$\left\{\begin{array}{l}2x-y=-1\\ x+\frac{1}{2}y=\frac{3}{2}\end{array}\right.$

Short answer

The solution of the system is$x=\frac{1}{2},y=2$,or, using ordered pair,$\left(\frac{1}{2},2\right)$.

Step-by-step solution

Step 1. Given information.

The given system of equation are

$\left\{\begin{array}{l}2x-y=-1\\ x+\frac{1}{2}y=\frac{3}{2}\end{array}\right.$

Step 2. Calculation.

The augmented matrix of the system is:

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

Perform the row operations $R_2=2r_2-r_1$:

$\left[\begin{array}{cc}\begin{array}{cc}2& -1\\ 0& 2\end{array}& \begin{array}{c}-1\\ 4\end{array}\end{array}\right]$

Perform the row operations $R_2=\frac{1}{2}{r}_2$:

$\left[\begin{array}{cc}\begin{array}{cc}2& -1\\ 0& 1\end{array}& \begin{array}{c}-1\\ 2\end{array}\end{array}\right]$

Use the obtained matrix to write the system of equations.

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

Step 3. Solve the equation.

Solve the equations to find the solution set.

Substitute $y=2$into the equation.

$$\begin{gathered} 2x-y=-1 \\ 2x-2=-1 \\ 2x=-1+2 \\ 2x=1 \\ x=\frac{1}{2} \end{gathered}$$

Therefore, the solution of the system is $x=\frac{1}{2},y=2$or, using ordered pair, $\left(\frac{1}{2},2\right)$.

Step 4. Conclusion.

The solution of the system is $x=\frac{1}{2},y=2$or, using ordered pair,$\left(\frac{1}{2},2\right)$.