Q. 21 - Intermediate Algebra

Question

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

 $\left\{\begin{array}{l}x=-3y+4\\ 2x+6y=8\end{array}\right.$

Step-by-Step Solution

Answer

The solution of the system $\left\{\begin{array}{l}x=-3y+4\\ 2x+6y=8\end{array}\right.$ has infinity many solutions.It is found  by a graph as,

Step 1. Given information.

Consider the given system of linear equations,

$$\begin{gathered} x=-3y+4 \\ 2x+6y=8 \end{gathered}$$

Step 2. Solve the first equation for y.

Find slope and y-intercept of the first equation.

$$\begin{gathered} x=-3y+4 \\ -3y=-x+4 \\ y=\left(\frac{1}{3}\right)x-\frac{4}{3} \end{gathered}$$

Here the slope is m=$\frac{1}{3}$and the y-intercept is b=$-\frac{4}{3}$.

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

Find slope and y-intercept of the second equation.

$$\begin{gathered} 2x+6y=8 \\ 6y=-2y+8 \\ y=(-\frac{1}{3})x+\frac{4}{3} \end{gathered}$$

Here the slope is m=$-\frac{1}{3}$and, the y-intercept b=$\frac{4}{3}$.

Step 4. Graph both the lines.

The graph of both equations on the same plane is drawn below,

• The lines are the same.
• Since every point on the line makes both equations true, there are infinity-ordered pairs that make both equations true.
• Therefore, there are infinity many solutions to this system.