Question

1. choose the statement that correctly describes how many solutions there are for this system of equations. graph of two lines on a coordinate plane a infinitely many solutions because the slopes are equal and the y-intercepts are equal b exactly one solution because the slopes are equal but the y-intercepts are not equal c no solution because the slopes are equal and the y-intercepts are not equal d exactly one solution because the slopes are not equal

Explanation:

Step1: Analyze the lines' slopes and intercepts

The two lines in the graph are parallel (since they have the same slope). For a system of linear equations \(y = m_1x + b_1\) and \(y = m_2x + b_2\):

• If \(m_1 = m_2\) and \(b_1 = b_2\), the lines are coincident (infinitely many solutions).
• If \(m_1 = m_2\) and \(b_1

eq b_2\), the lines are parallel (no solution).

• If \(m_1

eq m_2\), the lines intersect (one solution).

From the graph, the two lines have the same slope (parallel) and different \(y\)-intercepts.

Step2: Evaluate each option

• Option A: Incorrect. \(y\)-intercepts are not equal.
• Option B: Incorrect. Equal slopes with different intercepts mean no solution, not one.
• Option C: Correct. Equal slopes and different \(y\)-intercepts (parallel lines) mean no solution.
• Option D: Incorrect. Slopes are equal, so they don't intersect at one point.

Answer:

C. No solution because the slopes are equal and the \(y\)-intercepts are NOT equal