Question

Without graphing, determine the number of solutions and then classify the system of equations.

$$\begin{gathered} y=-\frac{1}{2}x+5 \\ x+2y=10 \end{gathered}$$

Short answer

The number of solution is infinite and the system of equations is coincident lines, consistent and dependent.

Step-by-step solution

Step 1. Given information

The system of equations is$\left\{\begin{array}{l}y=-\frac{1}{2}x+5\\ x+2y=10\end{array}\right.$

Step 2. Solve the first equation for y and find the slope and y-intercept. 

$y=-\frac{1}{2}x+5$

Here the slope is $-\frac{1}{2}$

And the $y$-intercept is$5$

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

$$\begin{gathered} x+2y=10 \\ 2y=10-x \\ y=-\frac{1}{2}x+5 \end{gathered}$$

Here the slope is$-\frac{1}{2}$

And the$y$-intercept is$5$

Step 4. Required answer

Here the slopes and $y$-intercepts are same.

Therefore, the system of equation is coincident lines with infinite solution and is consistent and dependent.