Question

system of linear equations shown below, classify the system as \consistent dependent,\ \consistent
istent.\ then, choose the best description of its solution. if the system has exactly one solution, giv

system a
1: $y = \frac{5}{2}x - 2$
2: $y = \frac{5}{2}x + 2$

graph of two parallel lines

system of equations is:
\bigcirc consistent dependent
\bigcirc consistent independent
\bigcirc inconsistent

means the system has:
\bigcirc a unique solution
solution: (\square, \square)
\bigcirc no solution
\bigcirc infinitely many solutions

system b
line 1: $y = -\frac{5}{2}x - 1$
line 2: $5x + 2y = -2$

graph of two coinciding lines

this system of equations is:
\bigcirc consistent dependent
\bigcirc consistent independent
\bigcirc inconsistent

this means the system has:
\bigcirc a unique solution
solution: (\square, \square)
\bigcirc no solution
\bigcirc infinitely many solutions

system c
line 1: $y = -x + 3$
line 2: $y = x + 1$

graph of two intersecting lines

this system of equations
\bigcirc consistent dependent
\bigcirc consistent independent
\bigcirc inconsistent

this means the system has
\bigcirc a unique solution
solution: (\square, \square)
\bigcirc no solution
\bigcirc infinitely many solutions

Explanation:

Step1: Analyze System A

The two lines in System A have the same slope ($\frac{5}{2}$) but different y-intercepts (-2 and +2). Parallel lines (same slope, different intercepts) never intersect, so the system has no solution. An inconsistent system has no solution, and parallel lines with different intercepts are inconsistent.

Step2: Analyze System B

First, rewrite Line 2 of System B: $5x + 2y = -2$ in slope-intercept form. Solve for $y$: $2y = -5x - 2$ → $y = -\frac{5}{2}x - 1$. This is the same as Line 1 ($y = -\frac{5}{2}x - 1$). So the two lines are coincident (same line), meaning there are infinitely many solutions. A consistent dependent system has infinitely many solutions (same line).

Step3: Analyze System C

Line 1: $y = -x + 3$ (slope -1), Line 2: $y = x + 1$ (slope 1). Different slopes mean the lines intersect at exactly one point. A consistent independent system has exactly one (unique) solution.

System A:

• Classification: inconsistent (no solution, parallel lines)
• Solution: no solution

System B:

• Classification: consistent dependent (infinitely many solutions, same line)
• Solution: infinitely many solutions

System C:

• Classification: consistent independent (unique solution, intersecting lines)
• Solution: unique solution (find intersection: set $-x + 3 = x + 1$ → $2x = 2$ → $x = 1$, then $y = -1 + 3 = 2$ or $y = 1 + 1 = 2$. So solution (1, 2))

Answer:

System A:

• Classification: inconsistent
• Solution: no solution

System B:

• Classification: consistent dependent
• Solution: infinitely many solutions

System C:

• Classification: consistent independent
• Solution: a unique solution, Solution: (1, 2)