Question

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.

system a
line 1: ( y = x + 4 )
line 2: ( y = -x )
graph of two lines intersecting
this system of equations is:
\\( \circ \\) consistent independent
\\( \circ \\) consistent dependent
\\( \circ \\) inconsistent
this means the system has:
\\( \circ \\) a unique solution
\\( \circ \\) infinitely many solutions
\\( \circ \\) no solution
solution: ( , )

system b
line 1: ( y = 2x )
line 2: ( y = 2x + 4 )
graph of two parallel lines
this system of equations is:
\\( \circ \\) consistent independent
\\( \circ \\) consistent dependent
\\( \circ \\) inconsistent
this means the system has:
\\( \circ \\) a unique solution
\\( \circ \\) infinitely many solutions
\\( \circ \\) no solution
solution: ( , )

system c
line 1: ( y = -x - 1 )
line 2: ( x + y = -1 )
graph of two coinciding lines
this system of equations is:
\\( \circ \\) consistent independent
\\( \circ \\) consistent dependent
\\( \circ \\) inconsistent
this means the system has:
\\( \circ \\) a unique solution
\\( \circ \\) infinitely many solutions
\\( \circ \\) no solution
solution: ( , )

Explanation:

System A:

Step1: Analyze slopes/intercepts

Line 1: \( y = x + 4 \) (slope \( m_1 = 1 \), y - intercept \( b_1 = 4 \))
Line 2: \( y = -x \) (slope \( m_2 = -1 \), y - intercept \( b_2 = 0 \))
Slopes are different (\( m_1
eq m_2 \)), so lines intersect at one point.

Step2: Classify system

Consistent independent (intersecting lines, one solution).

Step3: Find solution

Set \( x + 4 = -x \)
\( 2x = -4 \)
\( x = -2 \)
Substitute \( x = -2 \) into \( y = -x \): \( y = 2 \)
Solution: \( (-2, 2) \)

System B:

Step1: Analyze slopes/intercepts

Line 1: \( y = 2x \) (slope \( m_1 = 2 \), y - intercept \( b_1 = 0 \))
Line 2: \( y = 2x + 4 \) (slope \( m_2 = 2 \), y - intercept \( b_2 = 4 \))
Slopes equal (\( m_1 = m_2 \)), y - intercepts different (\( b_1
eq b_2 \)) → parallel lines.

Step2: Classify system

Inconsistent (parallel lines, no solution).

System C:

Step1: Rewrite Line 2

Line 2: \( x + y = -1 \) → \( y = -x - 1 \)
Line 1: \( y = -x - 1 \) (same as rewritten Line 2)

Step2: Classify system

Consistent dependent (same line, infinitely many solutions).

Answer:

System A:

• This system of equations is: consistent independent
• This means the system has: a unique solution
• Solution: \((-2, 2)\)

System B:

• This system of equations is: inconsistent
• This means the system has: no solution
• Solution: no solution

System C:

• This system of equations is: consistent dependent
• This means the system has: infinitely many solutions
• Solution: infinitely many solutions