Question

system a
line 1: $y = -\frac{2}{3}x - 4$
line 2: $y = -\frac{2}{3}x + 1$
graph of two lines
this system of equations is:
\bigcirc inconsistent
\bigcirc consistent dependent
\bigcirc consistent independent
this means the system has:
\bigcirc a unique solution
solution: (\square, \square)
\bigcirc no solution
\bigcirc infinitely many solutions

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

system c
line 1: $y = -\frac{3}{2}x$
line 2: $3x + 2y = 0$
graph of two lines coinciding
this system of equations is:
\bigcirc inconsistent
\bigcirc consistent dependent
\bigcirc consistent independent
this means the system has:
\bigcirc a unique solution
solution: (\square, \square)
\bigcirc no solution
\bigcirc infinitely many solutions

Explanation:

System A

Step1: Analyze slopes of lines

Line 1: \( y = -\frac{2}{3}x - 4 \), slope \( m_1 = -\frac{2}{3} \)
Line 2: \( y = -\frac{2}{3}x + 1 \), slope \( m_2 = -\frac{2}{3} \)
Since \( m_1 = m_2 \) and y - intercepts (\( -4 \) vs \( 1 \)) are different, lines are parallel.

Step2: Determine system type

Parallel lines never intersect, so the system has no solution. A system with no solution is inconsistent.

Step1: Analyze slopes of lines

Line 1: \( y = \frac{1}{2}x + 1 \), slope \( m_1 = \frac{1}{2} \)
Line 2: \( y = -2x + 1 \), slope \( m_2 = -2 \)
Since \( m_1
eq m_2 \), lines are not parallel (and not coincident), so they intersect at one point.

Step2: Determine system type

A system with one unique solution is consistent independent.

Step3: Find the solution

Set \( \frac{1}{2}x + 1 = -2x + 1 \)
\( \frac{1}{2}x + 2x = 1 - 1 \)
\( \frac{5}{2}x = 0 \)
\( x = 0 \)
Substitute \( x = 0 \) into \( y = \frac{1}{2}x + 1 \): \( y = 1 \)

Step1: Rewrite Line 2 in slope - intercept form

Line 2: \( 3x + 2y = 0 \)
Solve for \( y \): \( 2y = -3x \) → \( y = -\frac{3}{2}x \)

Line 1: \( y = -\frac{3}{2}x \)
So, Line 1 and Line 2 are the same line (same slope and y - intercept, y - intercept is \( 0 \) for both).

Step2: Determine system type

A system with coincident lines has infinitely many solutions, so it is consistent dependent.

Answer:

This system of equations is: Inconsistent
This means the system has: no solution

System B