Question

system b
line 1: $y = -\frac{1}{2}x+\frac{5}{2}$
line 2: $y = x + 4$
this system of equations is:
consistent independent
consistent dependent
inconsistent
this means the system has:
a unique solution
solution: (, )
no solution
infinitely many solutions
system c
line 1: $y = \frac{5}{2}x - 1$
line 2: $-5x + 2y=-2$
this system of equations is:
consistent independent
consistent dependent
inconsistent
this means the system has:
a unique solution
solution: (, )
no solution
infinitely many solutions

Explanation:

Step1: Analyze System B

For System B with equations $y =-\frac{1}{2}x+\frac{5}{2}$ and $y=x + 4$. Since the slopes $-\frac{1}{2}$ and $1$ are different, the lines intersect.

Step2: Solve System B

Set $-\frac{1}{2}x+\frac{5}{2}=x + 4$. Multiply through by 2 to get $-x + 5=2x+8$. Rearrange: $-x-2x=8 - 5$, so $-3x=3$, then $x=-1$. Substitute $x = - 1$ into $y=x + 4$, we get $y=-1 + 4=3$.

Step3: Analyze System C

For System C, rewrite $-5x+2y=-2$ as $y=\frac{5}{2}x-1$, which is the same as the first - equation $y=\frac{5}{2}x-1$. So the two equations represent the same line.

Step4: Determine consistency and solutions for System C

Since the two equations in System C are equivalent, it is consistent dependent and has infinitely many solutions.

Answer:

System B:

• This system of equations is: consistent independent
• This means the system has: a unique solution
• Solution: $(-1,3)$

System C:

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