Question

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

system a
1: $y = \frac{1}{2}x + 1$
2: $y = \frac{1}{2}x - 3$
graph of two parallel lines l1 and l2
system of equations is:
\\(\circ\\) inconsistent
\\(\circ\\) consistent dependent
\\(\circ\\) consistent independent
means the system has:
\\(\circ\\) a unique solution
solution: (\\(\square\\), \\(\square\\))
\\(\circ\\) infinitely many solutions
\\(\circ\\) no solution

system b
line 1: $y = x - 1$
line 2: $y = -x - 5$
graph of two intersecting lines l1 and l2
this system of equations is:
\\(\circ\\) inconsistent
\\(\circ\\) consistent dependent
\\(\circ\\) consistent independent
this means the system has:
\\(\circ\\) a unique solution
solution: (\\(\square\\), \\(\square\\))
\\(\circ\\) infinitely many solutions
\\(\circ\\) no solution

system c
line 1: $y = -\frac{1}{2}x + 1$
line 2: $x + 2y = 2$
graph of two lines (possibly coinciding or parallel)
this system of equations
\\(\circ\\) inconsistent
\\(\circ\\) consistent depen
\\(\circ\\) consistent indepe
this means the system ha
\\(\circ\\) a unique solution
solution: (\\(\square\\), \\(\square\\))
\\(\circ\\) infinitely many s
\\(\circ\\) no solution

Explanation:

Step1: Analyze System A

The two lines in System A have the same slope ($\frac{1}{2}$) but different y - intercepts (1 and - 3). Parallel lines (same slope, different y - intercepts) do not intersect, so the system has no solution. A system with no solution is inconsistent.

Step2: Analyze System B

The slopes of the two lines in System B are 1 (for $y = x - 1$) and - 1 (for $y=-x - 5$). Since the slopes are not equal, the lines are not parallel and will intersect at exactly one point. A system with exactly one solution is consistent independent, and it has a unique solution. To find the solution, we set $x - 1=-x - 5$. Solving for $x$: $x+x=-5 + 1\Rightarrow2x=-4\Rightarrow x=-2$. Substitute $x = - 2$ into $y=x - 1$, we get $y=-2 - 1=-3$. So the solution is $(-2,-3)$.

Step3: Analyze System C

First, rewrite Line 2: $x + 2y=2$ in slope - intercept form. Solve for $y$: $2y=-x + 2\Rightarrow y=-\frac{1}{2}x + 1$. This is the same as Line 1: $y =-\frac{1}{2}x+1$. So the two lines are identical (same slope and same y - intercept), which means the system has infinitely many solutions. A system with infinitely many solutions is consistent dependent.

Answer:

System A:

• System type: inconsistent
• Solution description: no solution

System B:

• System type: consistent independent
• Solution description: a unique solution
• Solution: $(-2,-3)$

System C:

• System type: consistent dependent
• Solution description: infinitely many solutions