Question

assignment: 4.3a parallel lines
determine which of the lines, if any, are parallel.
line a passes through $(-1, -2)$ and $(1, 0)$.
line b passes through $(4, 2)$ and $(2, -2)$.
line c passes through $(0, 2)$ and $(-1, 1)$.
line a passes through $(-1, 3)$ and $(1, 9)$.
line b passes through $(-2, 12)$ and $(-1, 14)$.
line c passes through $(3, 8.5)$ and $(6, 10.5)$.
find the equation of the line that passes through the given point and is parallel to the given line.

1. $(4, 1)$; $y = -3x + 2$
2. $(1, 2)$; $y = -5x + 4$

Explanation:

Step1: Calculate slope of Line a

Use slope formula $m=\frac{y_2-y_1}{x_2-x_1}$ with points $(-1,-2),(1,0)$:
$m_a=\frac{0-(-2)}{1-(-1)}=\frac{2}{2}=1$

Step2: Calculate slope of Line b

Use slope formula with points $(4,2),(2,-2)$:
$m_b=\frac{-2-2}{2-4}=\frac{-4}{-2}=2$

Step3: Calculate slope of Line c

Use slope formula with points $(0,2),(-1,1)$:
$m_c=\frac{1-2}{-1-0}=\frac{-1}{-1}=1$

Step4: Match equal slopes for parallelism

Parallel lines have equal slopes. $m_a=m_c=1$, $m_b=2$ is unique.
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Step5: Calculate slope of Line a (second set)

Use slope formula with points $(-1,3),(1,9)$:
$m_a=\frac{9-3}{1-(-1)}=\frac{6}{2}=3$

Step6: Calculate slope of Line b (second set)

Use slope formula with points $(-2,12),(-1,14)$:
$m_b=\frac{14-12}{-1-(-2)}=\frac{2}{1}=2$

Step7: Calculate slope of Line c (second set)

Use slope formula with points $(3,8.5),(6,10.5)$:
$m_c=\frac{10.5-8.5}{6-3}=\frac{2}{3}$

Step8: Check for equal slopes

All slopes are distinct, so no parallel lines here.
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Step9: Find parallel line for $y=-3x+2$

Parallel lines have same slope $m=-3$. Use point-slope form $y-y_1=m(x-x_1)$ (missing point assumed to be $(x_1,y_1)$; if we assume a common missing point like $(0,0)$ for example):
$y-0=-3(x-0) \implies y=-3x$
Note: If a specific point was intended but cut off, substitute that $(x_1,y_1)$ into the formula.

Step10: Find parallel line for $(1,2); y=-5x+4$

Parallel slope $m=-5$. Use point-slope form:
$y-2=-5(x-1)$
Simplify: $y-2=-5x+5 \implies y=-5x+7$

Answer:

1. For the first set of lines: Line a and Line c are parallel.
2. For the second set of lines: No lines are parallel.
3. For the line parallel to $y=-3x+2$ (assuming point $(0,0)$): $y=-3x$ (substitute the correct given point if available for an exact answer)
4. For the line through $(1,2)$ parallel to $y=-5x+4$: $y=-5x+7$