Question

which statements about the system are true? check all that apply.
the system has one solution.
the system graphs parallel lines.
both lines have the same slope.
both lines have the same ( y )-intercept.
the equations graph the same line.
the solution is the intersection of the 2 lines.

Explanation:

To solve this, we analyze each statement based on linear system properties:

Step 1: Recall Linear System Basics

A linear system \(

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

\) has:

• One solution if \( m_1

eq m_2 \) (lines intersect once).

• No solution if \( m_1 = m_2 \) and \( b_1

eq b_2 \) (parallel lines).

• Infinitely many solutions if \( m_1 = m_2 \) and \( b_1 = b_2 \) (same line).

Analyze Each Statement:

1. "The system has one solution."

True if lines have different slopes (intersect once).

1. "The system graphs parallel lines."

True only if \( m_1 = m_2 \) and \( b_1
eq b_2 \) (no info on equations, but if we assume a typical "consistent" vs "inconsistent" context, this is false unless slopes are equal and y - intercepts differ).

1. "Both lines have the same slope."

True if \( m_1 = m_2 \) (parallel or same line).

1. "Both lines have the same y - intercept."

True only if \( b_1 = b_2 \) (same line).

1. "The equations graph the same line."

True only if \( m_1 = m_2 \) and \( b_1 = b_2 \) (infinitely many solutions).

1. "The solution is the intersection of the 2 lines."

Always true for a linear system (solution is where lines meet, even if infinitely many or one).

Key Deductions (Assuming a Typical System with One Solution):

• If the system has one solution, lines have different slopes (so "same slope" is false), but:
• "The system has one solution" is true (lines intersect once).
• "The solution is the intersection of the 2 lines" is always true (by definition of solution for linear systems).

Final True Statements:

• The system has one solution (if slopes differ).
• The solution is the intersection of the 2 lines (always true for linear systems).

• "The system has one solution" is true if lines intersect (different slopes).
• "The solution is the intersection of the 2 lines" is always true (a solution to a linear system is where the lines meet).
• Other statements depend on slope/y - intercept equality, which is not confirmed by the problem’s context (no equations given, but by definition of linear system solutions, the intersection is the solution).

Answer:

• The system has one solution.
• The solution is the intersection of the 2 lines.