Q 16. - Intermediate Algebra

Question

In the following exercises,solve the following system of equations by graphing.

$\left\{\begin{array}{l}-x+2y=-6\\ y=-\frac{1}{2}x-1\end{array}\right.$

Step-by-Step Solution

Answer

The solution of the system of the linear equation (2,-2).

Step 1. Given information.

Given equations,

$\begin{array}{rcl}-x+2y& =& -6\\ y& =& -\frac{1}{2}x-1\end{array}$

Step 2. Solve the first equation for y.

Solve the equation and find the slope and y-intercept.

$$\begin{gathered} -x+2y=-6 \\ 2y=x-6 \\ y=\frac{1}{2}x-3 \end{gathered}$$

Here the slope is $m=\frac{1}{2}$and, the y-intercept is $b=-3$.

Step 3. Solve the second equation for y , and find the slope and y -intercept.

Find the slope and intercept.

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

Here the slope is m=$-\frac{1}{2}$and, the y-intercept is b=$-1$.

Step 4. Graph both the lines.

The graph of the both lines is drown below,

Step 5.checking

First substitute x = 2, y = -2 into the equation $-x+2y=-6$.

$\begin{array}{rcl}-x+2y& =& -6\\ -2+2(-2)& =& -6\\ -2-4& =& -6\\ -6& =& -6\end{array}$

This is true.

Also,Substitute x=2,y=-2 into the equation $y=-\frac{1}{2}x-1$.

$\begin{array}{rcl}y& =& -\frac{1}{2}x-1\\ -2& =& -\frac{1}{2}\left(2\right)-1\\ -2& =& -1-1\\ -2& =& -2\end{array}$

This is also true.

Therefore, the solution of the system of the linear equation (2,-2)..