Question

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

$$\begin{cases} y = -2x + 3 \\\\ y = 3x - 2 \\end{cases}$$

one solution an infinite number of solutions no solutions

Explanation:

Step1: Analyze the 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 equation \(y=-2x + 3\), the slope \(m_1=-2\), and for the equation \(y = 3x-2\), the slope \(m_2 = 3\). Since \(m_1
eq m_2\), the two lines are not parallel.

Step2: Determine the number of solutions

Two non - parallel lines in a plane (the coordinate plane, which is a two - dimensional space) intersect at exactly one point. Each point of intersection of the two lines represents a solution to the system of equations. So, a system of two linear equations with non - parallel lines (different slopes) has exactly one solution.

Answer:

one solution