Question

Solve the system by graphing: $\left\{\begin{array}{l}y=\frac{1}{2}x-4\\ 2x-4y=16\end{array}\right.$

Short answer

The system of equations$\left\{\begin{array}{l}y=\frac{1}{2}x-4\\ 2x-4y=16\end{array}\right.$has infinitely many solutions and it can be seen in the graph as

Step-by-step solution

Step 1. Given Information.

We are given a system of equations $\left\{\begin{array}{l}y=\frac{1}{2}x-4\\ 2x-4y=16\end{array}\right.$and we need to find its solution by graphing.

Step 2. Graph the first equation.

The equation $y=\frac{1}{2}x-4$is in slope-intercept form, so its slope is $\frac{1}{2}$and its $y$intercept is $-4$.

So its graph is given as,

Step 3. Graph the second equation on the same rectangular coordinate system.

Solving the equation $2x-4y=16$for $y$we get,

localid="1644388026323" $\begin{array}{rcl}2x-4y-2x& =& 16-2x\\ -4y& =& -2x+16\\ \frac{-4y}{-4}& =& \frac{-2x+16}{-4}\\ y& =& \frac{1}{2}x-4\end{array}$

So the slope of the line is $\frac{1}{2}$and its $y$intercept is $-4$.

So its graph can be drawn on the same plane as,

Step 4. Identify the solution to the system.

• From the graph, it can be seen that the graph of the equations is the same.
• Since every point on the line makes both equations true, there are infinitely many ordered pairs that make both equations true.
• There are infinitely many solutions to this system.