Question

1. use patterns and structure for each graph, draw a line to make a system of equations with the given number of solutions.

a. no solution
b. one solution

Explanation:

Step1: Analyze Part a (No Solution)

For a system of linear equations to have no solution, the two lines must be parallel (same slope, different y - intercepts). The given line has a slope. Let's assume the given line has a slope \(m\). To draw a line with no solution, we draw a line with the same slope as the given line but a different y - intercept. For example, if the given line is \(y = x\) (from the graph, passing through the origin with slope 1), we can draw a line like \(y=x + 2\) (same slope 1, different y - intercept 2).

Step2: Analyze Part b (One Solution)

For a system of linear equations to have one solution, the two lines must intersect at exactly one point, which means they have different slopes. The given line has a slope \(m_1\). We draw a line with a slope \(m_2
eq m_1\). For example, if the given line is \(y = x\) (slope 1), we can draw a line like \(y=-x\) (slope - 1), which will intersect the given line at the origin (one solution).

Answer:

Part a (No Solution)

Draw a line parallel to the given line (same slope, different y - intercept). For example, if the given line is \(y = x\), draw \(y=x + 2\) (or any line with slope 1 and \(y\) - intercept \(
eq0\)).

Part b (One Solution)

Draw a line with a different slope than the given line. For example, if the given line is \(y = x\) (slope 1), draw \(y=-x\) (slope - 1) which intersects the given line at one point.