Question

if a system of linear equations has one solution, what does it mean about the two lines when graphed?
intersecting lines - different slope (y - intercept can be the same)
parallel lines - same slope, different y - intercept
the same line - same slope, same y - intercept

Explanation:

To determine the relationship between two lines in a system of linear equations with one solution, we analyze the properties of linear equations (in the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the \(y\)-intercept):

Step 1: Recall the conditions for the number of solutions

• One solution: The lines intersect at a single point. For two lines \(y = m_1x + b_1\) and \(y = m_2x + b_2\) to intersect at one point, their slopes must be different (\(m_1

eq m_2\)) (if slopes were equal, the lines would be parallel or coincident, not intersecting at one point). The \(y\)-intercepts can be the same or different, but the key is the slopes are different.

• No solution (parallel lines): Same slope (\(m_1 = m_2\)) and different \(y\)-intercepts (\(b_1

eq b_2\)).

• Infinitely many solutions (coincident lines): Same slope (\(m_1 = m_2\)) and same \(y\)-intercept (\(b_1 = b_2\)).

Step 2: Match with the given options

• The green option states: "Intersecting lines - different slope (y-intercept can be the same)" → This matches the condition for one solution (intersecting lines have different slopes, \(y\)-intercepts can vary).
• The blue option ("Parallel lines - same slope, different \(y\)-intercept") describes no solution.
• The orange option ("The same line - same slope, same \(y\)-intercept") describes infinitely many solutions.

Answer:

Intersecting lines - different slope (y - intercept can be the same)