Question

for each system of linear equations shown below, classify the system as \consistent dependent,\ \consistent independent,\ or \inconsistent.\ then, choose the best description of its solution. if the system has exactly one solution, give its solution.
your answer
system a
line 1: $y = -x - 1$
line 2: $y = x - 5$
graph for system a
system b
line 1: $y = \frac{3}{4}x + 1$
line 2: $-3x + 4y = 4$
graph for system b
system c
line 1: $y = -x + 4$
line 2: $y = -x - 4$
graph for system c

Explanation:

System A:

Step 1: Analyze slopes

Line 1: \( y = -x - 1 \), slope \( m_1 = -1 \)
Line 2: \( y = x - 5 \), slope \( m_2 = 1 \)
Since \( m_1
eq m_2 \), lines intersect at one point.

Step 2: Solve the system

Set \( -x - 1 = x - 5 \)
\( -x - x = -5 + 1 \)
\( -2x = -4 \)
\( x = 2 \)
Substitute \( x = 2 \) into \( y = x - 5 \): \( y = 2 - 5 = -3 \)
Solution: \( (2, -3) \)
Classification: Consistent Independent (one solution)

System B:

Step 1: Rewrite Line 2

Line 2: \( -3x + 4y = 4 \) → \( 4y = 3x + 4 \) → \( y = \frac{3}{4}x + 1 \)
Line 1: \( y = \frac{3}{4}x + 1 \)
Same equation, so lines are coincident.

Step 2: Classification

Consistent Dependent (infinitely many solutions)

System C:

Step 1: Analyze slopes

Line 1: \( y = -x + 4 \), slope \( m_1 = -1 \)
Line 2: \( y = -x - 4 \), slope \( m_2 = -1 \)
Check y-intercepts: \( b_1 = 4 \), \( b_2 = -4 \) (different), so parallel lines.

Step 2: Classification

Inconsistent (no solution)

Answer:

• System A: Consistent Independent, Solution \((2, -3)\)
• System B: Consistent Dependent
• System C: Inconsistent