CPM Homework Help : CC3 Problem 9-63

Determine by inspection whether the lines in each system below intersect, coincide, or are parallel. Do not actually solve the systems. Justify your reasons.

Hint:

Parallel lines have equal slope.
Lines that coincide have equal slope and equal $y$ -intercepts.
Lines that neither coincide or are parallel will intersect once.

$y = 2x + 3\\y=\frac { 1 } { 2 }x − 2$
Hint (a):
The coefficient of $x$ tells you the slope.
Are they the same?

$2x + 3y = 6 \\2x + 3y = 9$
Hint (b):
These equations have the same coefficient for $x$ and $y$ , but have different constants. What does this tell you about that lines?

$y=\frac { 1 } { 3 }x + 2\\ y=\frac { 1 } { 3 }x - 2$
Hint (c):
Notice that the coefficients of $x$ are the same in the two equations.
Answer (c):
The lines are parallel.

$\:\:\:\:\:\:x - 2y = 4 \\-2x + 4y = -8$
Hint (d):
Write both equations in $y = mx + b$ form.
Step 1 (d):
$x − 2y = 4$
${\it y} = \frac{1}{2}{\it x} - 2$
Step 2 (d):
$−2x + 4y = −8$
${\it y} = \frac{1}{2}{\it x} - 2$
Answer (d):
Because the two equations are equal, the lines coincide.