Q 11 - Intermediate Algebra

Question

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

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

Step-by-Step Solution

Answer

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

Step 1. Solving equations for the intersection points

First, solve both of these equations for y such that their slopes and y intercepts may be easily graphed.

And find the slope and y-intercept by solving the first equation for y.

$$\begin{gathered} x + y = - 4 \\ y = - x - 4 \\ \mathrm{Here} \mathrm{the} \mathrm{slope} \mathrm{is} m = - 1 \\ \mathrm{And} \mathrm{the} \mathrm{y}-\mathrm{intercept} \mathrm{is} b = - 4 \end{gathered}$$

Again find the slope and y-intercept by solving the Second equation for y. 

$$\begin{gathered} - x + 2y = - 2 \\ 2y = x - 2 \\ y = \frac{1}{2}\times x -\frac{2}{2} \\ y =\frac{1}{2} \times x - 1 \\ \mathrm{Here} \mathrm{the} \mathrm{slope} \mathrm{is} m =\frac{1}{2} \\ \mathrm{And} \mathrm{the} \mathrm{y}-\mathrm{intercept} \mathrm{is} b = - 1 \end{gathered}$$

Step 2. Graph obtained

Step 3. Checking the equations

$$\begin{gathered} \mathrm{Check}: \\ \mathrm{First} \mathrm{substitute} x = -2, \\ y=-2 \mathrm{into} \mathrm{the} \mathrm{equation} \\ x+y=-4 \\ -2-2=-4 \\ -4=4 \\ \mathrm{This} \mathrm{is} \mathrm{true}. \end{gathered}$$

$$\begin{gathered} \mathrm{Also} \mathrm{substitute} x = - 2 , \\ y = - 2 \mathrm{into} \mathrm{the} \mathrm{equation} \\ - x + 2y = - 2 \\ - x + 2y = - 2 \\ - (- 2) + 2(- 2) = - 2 \\ 2-4=-2 \\ -2=-2 \\ \mathrm{This} \mathrm{is} \mathrm{true}. \end{gathered}$$

$$\begin{gathered} \mathrm{Also} \mathrm{substitute} x = - 2 , \\ y = - 2 \mathrm{into} \mathrm{the} \mathrm{equation} \\ - x + 2y = - 2 \\ - x + 2y = - 2 \\ - (- 2) + 2(- 2) = - 2 \\ 2-4=-2 \\ -2=-2 \\ \mathrm{This} \mathrm{is} \mathrm{true}. \end{gathered}$$