Question

system a
line 1: $y = x$
line 2: $-x + y = 0$
this system of equations is:

system b
line 1: $y = x - 3$
line 2: $y = \frac{1}{2}x - \frac{5}{2}$
this system of equations is:

system c
line 1: $y = \frac{2}{3}x$
line 2: $y = \frac{2}{3}x + 3$
this system of equations is:

Explanation:

Step1: Analyze System A

Rewrite Line 2: $-x + y = 0$ can be rearranged to $y = x$, which is identical to Line 1. This means the two lines are the same, so there are infinitely many solutions. A system with infinitely many solutions is consistent dependent.

Step2: Analyze System B

Compare slopes: Line 1 has slope $m_1=1$, Line 2 has slope $m_2=\frac{1}{2}$. Since $m_1
eq m_2$, the lines intersect at exactly one point, so there is one unique solution. A system with one unique solution is consistent independent.

Step3: Analyze System C

Compare slopes: Line 1 has slope $m_1=\frac{2}{3}$, Line 2 has slope $m_2=\frac{2}{3}$. The y-intercepts are $b_1=0$ and $b_2=3$, so the lines are parallel and never intersect. A system with no solutions is inconsistent.

Answer:

• System A: consistent dependent
• System B: consistent independent
• System C: inconsistent