Question

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

$$\begin{cases} y = 2x - 1 \\\\ y = -\\frac{1}{2}x + 4 \\end{cases}$$

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

Explanation:

Step1: Analyze the slopes of the lines

The first equation \( y = 2x - 1 \) is in slope - intercept form \( y=mx + b \), where the slope \( m_1=2 \). The second equation \( y=-\frac{1}{2}x + 4 \) has a 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 \(

$$\begin{cases}y = m_1x + b_1\\y=m_2x + b_2\end{cases}$$

\), if the two lines are not parallel (i.e., \( m_1
eq m_2 \)), they will intersect at exactly one point. So the system of linear equations has one solution.

Answer:

one solution