Chapter 11.2, Problem 44AYU

In problems 39-74, solve each system of equations using matrices (row operations). If the system has no solution, say that it is inconsistent.

$\{\begin{array}{l}\frac{1}{2}x+y=-2\\ x-2y=8\end{array}$

To determine

To find: The solution to the given system of equations using matrices.

Answer to Problem 44AYU

Solution:

$x=2,y=-3$

Consistent system.

Explanation of Solution

Given:

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

$x-2y=8$

Formula used:

To solve a system of two equations in $x$ and $y$ using matrices:

Step 1: Write the corresponding matrix associated with the system of equations.

Step 2: Use elementary row operations to get equivalent matrix of the form:

$\left[\begin{array}{cc}1& a\\ 0& 1\end{array}|\begin{array}{c}b\\ c\end{array}\right]$; where $a,b,c$ are constants.

Step 3: Solve for $x$ and $y$.

Calculation:

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

$x-2y=8$

Rewrite the equations:

$2\left(\frac{1}{2}x+y\right)=2\left(-2\right)$

$x-2y=8$

Or

$x+2y=-4$

$x-2y=8$

The corresponding matrix associated with the above system of equations is:

$\left[\begin{array}{cc}1& 2\\ 1& -2\end{array}|\begin{array}{c}-4\\ 8\end{array}\right]$

The element in (row1, column1) position is already 1.

To get the (row2, column1) position as zero, multiply row1 by $-1$ and add to row2:

$\left[\begin{array}{cc}1& 2\\ -1\left(1\right)+1& -1\left(2\right)-2\end{array}|\begin{array}{c}-4\\ -1\left(-4\right)+8\end{array}\right]$

Simplify further:

$\left[\begin{array}{cc}1& 2\\ 0& -4\end{array}|\begin{array}{c}-4\\ 12\end{array}\right]$

Use elementary row operation to get the (row2, column2) position as 1.

Divide row2 by $-4$:

$\left[\begin{array}{cc}1& 2\\ \frac{0}{-4}& \frac{-4}{-4}\end{array}|\begin{array}{c}-4\\ \frac{12}{-4}\end{array}\right]$

Simplify further:

$\left[\begin{array}{cc}1& 2\\ 0& 1\end{array}|\begin{array}{c}-4\\ -3\end{array}\right]$

The above matrix corresponds to the system of equations:

$x+2y=-4$

$y=-3$

Substitute $y=-3$ in the first equation, $x+2y=-4$.

$x+2\left(-3\right)=-4$

$x-6=-4$

$x=-4+6$

$x=2$

Thus the solutions of the given system of equations are: $x=2;y=-3$.