Question

1. line 1: $y=-\frac{3}{2}x-4$

line 2: $5x+2y=-8$
this system of equations is:
○ consistent dependent
○ consistent independent
○ inconsistent
this means the system has:
○ a unique solution
○ no solution
solution: (□,□)

1. line 1: $y=-\frac{1}{2}x+4$

line 2: $y=-x+5$
this system of equations is:
○ consistent dependent
○ consistent independent
○ inconsistent
this means the system has:
○ a unique solution
○ no solution
solution: (□,□)

1. line 1: $y=x$

line 2: $y=x-2$
this system of equations is:
○ consistent dependent
○ consistent independent
○ inconsistent
this means the system has:
○ a unique solution
○ no solution
solution: (□,□)

Explanation:

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System 1 (Top):

Step1: Rewrite Line 2 to slope-intercept

Rearrange $5x+2y=-8$ to solve for $y$:
$2y=-5x-8$
$y=-\frac{5}{2}x-4$

Step2: Compare equations

Line 1: $y=-\frac{5}{2}x-4$; Line 2: $y=-\frac{5}{2}x-4$
The lines are identical (same slope, same intercept).

Step3: Classify system & solutions

Identical lines mean infinitely many solutions: consistent dependent, and the system has infinitely many solutions (not unique, not no solution).

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System 2 (Middle):

Step1: Identify slopes/intercepts

Line 1: $y=-\frac{1}{2}x+4$ (slope $-\frac{1}{2}$, intercept $4$)
Line 2: $y=-x+5$ (slope $-1$, intercept $5$)

Step2: Find intersection via substitution

Set $-\frac{1}{2}x+4=-x+5$
$\frac{1}{2}x=1$
$x=2$
Substitute $x=2$ into Line 1: $y=-\frac{1}{2}(2)+4=3$

Step3: Classify system & solutions

Distinct intersecting lines: consistent independent, with a unique solution $(2,3)$.

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System 3 (Bottom):

Step1: Compare slopes/intercepts

Line 1: $y=x$ (slope $1$, intercept $0$)
Line 2: $y=x-2$ (slope $1$, intercept $-2$)

Step2: Classify system & solutions

Same slope, different intercepts: lines are parallel, no intersection. System is inconsistent, with no solution.

Answer:

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1. Top System:

This system of equations is: consistent dependent
This means the system has: infinitely many solutions (not listed as an option, but the lines are identical)
Solution: All points on $y=-\frac{5}{2}x-4$

1. Middle System:

This system of equations is: consistent independent
This means the system has: a unique solution
Solution: $(2, 3)$

1. Bottom System:

This system of equations is: inconsistent
This means the system has: no solution
Solution: No solution