Q 13 - Intermediate Algebra

Question

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

$$\begin{gathered} -2x+3y=3 \\ x+3y=12 \end{gathered}$$

Step-by-Step Solution

Answer

The solution of the system of linear equations is (3,3).

Step 1. Given information.

Two linear equations are,

$$\begin{gathered} -2x+3y=3 \\ x+3y=12 \end{gathered}$$

Step 2. 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.

$$\begin{gathered} - 2x + 3y = 3 \\ 3y = 2x + 3 \\ y = \frac{2}{3} \times x +\frac{3}{3} \\ y =\frac{2}{3} \times x + 1 \\ \mathrm{Here} \mathrm{the} \mathrm{slope} \mathrm{is} m =\frac{2}{3} \\ \mathrm{And} \mathrm{the} \mathrm{y}-\mathrm{intercept} \mathrm{is} b = 1 \end{gathered}$$

$\begin{array}{rcl}-2x + 3y & =& 3 \\ 3y & =& 3+2x\\ y & =& \frac{3}{3} +\frac{2}{3}x \\ y & =& \frac{2}{3} \times x +1 \\ \mathrm{Here} \mathrm{the} \mathrm{slope} \mathrm{is} m & =& \frac{2}{3} \\ \mathrm{And} \mathrm{the} \mathrm{y}-\mathrm{intercept} \mathrm{is} b & =& 1\end{array}$

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

 

$$\begin{gathered} x + 3y = 12 \\ 3y = - x + 12 \\ y = - \frac{1}{3} \times x +\frac{12}{3} \\ y = - \frac{1}{3} \times x + 4 \\ \mathrm{Here} \mathrm{the} \mathrm{slope} \mathrm{is} m = - \frac{1}{3} \\ \mathrm{And} \mathrm{the} \mathrm{y}-\mathrm{intercept} \mathrm{is} b = 4 \end{gathered}$$

Step 3. Graph obtained.

Step 4. Checking the equations

$$\begin{gathered} \mathrm{First} \mathrm{substitute} x = 3 , \\ y = 3 \mathrm{into} \mathrm{the} \mathrm{equation} \\ -2 x + 3y = 3 \\ - 2x + 3y = 3 \\ -2\times 3+3\left(3\right)=3 \\ -6+9=3 \\ 3=3 \\ \mathrm{This} \mathrm{is} \mathrm{true}. \end{gathered}$$

$$\begin{gathered} \mathrm{First} \mathrm{substitute} x = 3 , \\ y = 3 \mathrm{into} \mathrm{the} \mathrm{equation} \\ x + 3y = 12 \\ x + 3y = 12 \\ 3+3\left(3\right)=12 \\ 3+9=12 \\ 12=12 \\ \mathrm{This} \mathrm{is} \mathrm{true}. \end{gathered}$$