Question

1. \

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

1. \

$$\begin{cases} y = -2x - 1 \\\\ 3x - 4y = -40 \\end{cases}$$

1. \

$$\begin{cases} 3x - 2y = -16 \\\\ x + y = -7 \\end{cases}$$

1. \

$$\begin{cases} 2x + y = 8 \\\\ x = 5 \\end{cases}$$

questions:

1. if a system of linear equations has one solution, what does this mean about the two lines?
2. if a system of linear equations has no solution, what does this mean about the two lines?
3. if a system of linear equations has infinitely many solutions, what does this mean about the two lines?

Explanation:

1)

A system of linear equations with one solution means the two lines intersect at exactly one point. This occurs when the lines have different slopes (for non - vertical/non - horizontal lines) or one is vertical and the other is horizontal (or has a non - undefined slope) and they cross at a single point. In terms of linear equations in two variables \(y = m_1x + b_1\) and \(y=m_2x + b_2\), if \(m_1
eq m_2\), the lines intersect at one point, giving one solution.

A system of linear equations with no solution means the two lines are parallel. Parallel lines have the same slope (\(m_1 = m_2\)) but different y - intercepts (\(b_1
eq b_2\)) for the slope - intercept form \(y = mx + b\). Since parallel lines never intersect, there are no points that satisfy both equations simultaneously.

A system of linear equations with infinitely many solutions means the two lines are coincident (they are the same line). This happens when one equation is a multiple of the other. For example, if we have \(y=2x + 3\) and \(2y = 4x+6\) (where the second equation is just 2 times the first), every point on one line is also on the other line, so there are infinitely many points that satisfy both equations.

Answer:

The two lines intersect at exactly one point.

2)