Question

how many solutions are there to this system of linear equations?\

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

\
no solutions\
one solution\
an infinite number of solutions

Explanation:

Step1: Analyze slopes of the lines

The two equations are in slope - intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y - intercept. For the first equation \(y=-2x + 5\), the slope \(m_1=-2\). For the second equation \(y =-\frac{1}{2}x+2\), the slope \(m_2 =-\frac{1}{2}\). Since \(m_1
eq m_2\), the two lines are not parallel.

Step2: Determine the number of solutions

For a system of linear equations in two variables, if the two lines are not parallel (i.e., their slopes are different), they will intersect at exactly one point. So the system of equations \(

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

\) has one solution.

Answer:

one solution