Question

solving systems by inspection
six systems of equations are given below. observe each system and complete the table by recording the number of solutions the system has and an explanation of your choice.
a
$y = \frac{1}{5}x - 2$
$-\frac{1}{5}x + y = -2$
b
$y = \frac{3}{2}x - 4$
$y = -\frac{3}{2}x - 4$
c
$-3x + 2y = 5$
$3x - 2y = 2$
d
$y = \frac{1}{2}x - 1$
$2y = x + 4$
e
$-3x + 9y = -12$
$-x + 3y = -4$
f
$5x + y = 8$
$-5x + y = -2$

system	# solutions	explain
b	infinite many	it is the same
c
d
e
f

©maneuvering the middle llc, 2018

Explanation:

Step1: Analyze System A

Rewrite second equation:

$$\begin{align} -\frac{1}{5}x + y &= -2 \\ y &= \frac{1}{5}x - 2 \end{align}$$

This matches the first equation, so they are the same line.

Step2: Analyze System B

First line: $y = \frac{3}{2}x - 4$; Second line: $y = -\frac{3}{2}x - 4$.
Different slopes ($\frac{3}{2}
eq -\frac{3}{2}$), so lines intersect once.

Step3: Analyze System C

Rewrite equations in slope-intercept form:

$$\begin{align} -3x + 2y &= 5 \implies y = \frac{3}{2}x + \frac{5}{2} \\ 3x - 2y &= 2 \implies y = \frac{3}{2}x - 1 \end{align}$$

Same slope, different y-intercepts: parallel lines, no intersection.

Step4: Analyze System D

Rewrite second equation:

$$\begin{align} 2y &= x + 4 \\ y &= \frac{1}{2}x + 2 \end{align}$$

First line: $y = \frac{1}{2}x - 1$. Same slope, different y-intercepts: parallel lines, no intersection.

Step5: Analyze System E

Simplify first equation by dividing by 3:

$$\begin{align} -3x + 9y &= -12 \implies -x + 3y = -4 \end{align}$$

This matches the second equation, so they are the same line.

Step6: Analyze System F

Rewrite equations in slope-intercept form:

$$\begin{align} 5x + y &= 8 \implies y = -5x + 8 \\ -5x + y &= -2 \implies y = 5x - 2 \end{align}$$

Different slopes ($-5
eq 5$), so lines intersect once.

Answer:

SYSTEM	# SOLUTIONS	EXPLAIN
B	1	The lines have different slopes, so they intersect at exactly one point.
C	0	The lines have the same slope but different y-intercepts (parallel, no overlap).
D	0	The lines have the same slope but different y-intercepts (parallel, no overlap).
E	Infinitely many	The two equations represent the same line.
F	1	The lines have different slopes, so they intersect at exactly one point.