Question

writing equations of parallel and perpendicular lines
determine whether the graphs of each pair of equations are parallel, perpendicular, coinciding, or none of these.

1. $x + 3y = 18$

$3x + 9y = 12$

1. $2x - 4y = 8$

$x - 2y = 4$

1. $-3x + 2y = 6$

$2x + 3y = 12$

1. $x + y = 6$

$3x - y = 6$

1. $4x + 8y = 2$

$2x + 4y = 8$

1. $3x - y = 9$

$6x - 2y = 18$

write the standard form of the equation of the line that is parallel to the graph of the given equation and that passes through the point with the given coordinates.

1. $2x + y - 5 = 0; (0, 4)$
2. $3x - y + 3 = 0; (-1, -2)$
3. $3x - 2y + 8 = 0; (2, 5)$

Explanation:

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Part 1: Classify Line Pairs

Step1: Convert to slope-intercept form ($y=mx+b$)

For each equation, solve for $y$ to find slope $m$.
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Pair 1: $x+3y=18$; $3x+9y=12$

Step1: Find slope of first line

$3y = -x + 18 \implies y = -\frac{1}{3}x + 6$, slope $m_1=-\frac{1}{3}$

Step2: Find slope of second line

$9y = -3x + 12 \implies y = -\frac{1}{3}x + \frac{4}{3}$, slope $m_2=-\frac{1}{3}$

Step3: Compare slopes and intercepts

$m_1=m_2$, intercepts $6
eq\frac{4}{3}$ → Parallel
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Pair 2: $2x-4y=8$; $x-2y=4$

Step1: Find slope of first line

$-4y = -2x + 8 \implies y = \frac{1}{2}x - 2$, slope $m_1=\frac{1}{2}$

Step2: Find slope of second line

$-2y = -x + 4 \implies y = \frac{1}{2}x - 2$, slope $m_2=\frac{1}{2}$

Step3: Compare slopes and intercepts

$m_1=m_2$, intercepts equal → Coinciding
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Pair 3: $-3x+2y=6$; $2x+3y=12$

Step1: Find slope of first line

$2y = 3x + 6 \implies y = \frac{3}{2}x + 3$, slope $m_1=\frac{3}{2}$

Step2: Find slope of second line

$3y = -2x + 12 \implies y = -\frac{2}{3}x + 4$, slope $m_2=-\frac{2}{3}$

Step3: Check perpendicularity

$m_1 \times m_2 = \frac{3}{2} \times -\frac{2}{3} = -1$ → Perpendicular
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Pair 4: $x+y=6$; $3x-y=6$

Step1: Find slope of first line

$y = -x + 6$, slope $m_1=-1$

Step2: Find slope of second line

$-y = -3x + 6 \implies y = 3x - 6$, slope $m_2=3$

Step3: Compare slopes

$m_1
eq m_2$, $m_1 \times m_2
eq -1$ → None of these
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Pair 5: $4x+8y=2$; $2x+4y=8$

Step1: Find slope of first line

$8y = -4x + 2 \implies y = -\frac{1}{2}x + \frac{1}{4}$, slope $m_1=-\frac{1}{2}$

Step2: Find slope of second line

$4y = -2x + 8 \implies y = -\frac{1}{2}x + 2$, slope $m_2=-\frac{1}{2}$

Step3: Compare slopes and intercepts

$m_1=m_2$, intercepts $\frac{1}{4}
eq2$ → Parallel
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Pair 6: $3x-y=9$; $6x-2y=18$

Step1: Find slope of first line

$-y = -3x + 9 \implies y = 3x - 9$, slope $m_1=3$

Step2: Find slope of second line

$-2y = -6x + 18 \implies y = 3x - 9$, slope $m_2=3$

Step3: Compare slopes and intercepts

$m_1=m_2$, intercepts equal → Coinciding
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Part 2: Write Parallel Line Equations (Standard Form $Ax+By=C$)

Parallel lines have equal slopes. Use point-slope form $y-y_1=m(x-x_1)$, then convert to standard form.
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Problem 7: $2x+y-5=0$; $(0,4)$

Step1: Find slope of given line

$y = -2x + 5$, slope $m=-2$

Step2: Use point-slope form

$y - 4 = -2(x - 0)$

Step3: Convert to standard form

$y - 4 = -2x \implies 2x + y = 4$
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Problem 8: $3x-y+3=0$; $(-1,-2)$

Step1: Find slope of given line

$-y = -3x - 3 \implies y = 3x + 3$, slope $m=3$

Step2: Use point-slope form

$y - (-2) = 3(x - (-1))$

Step3: Simplify and convert to standard form

$y + 2 = 3x + 3 \implies 3x - y = -1$
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Problem 9: $3x-2y+8=0$; $(2,5)$

Step1: Find slope of given line

$-2y = -3x - 8 \implies y = \frac{3}{2}x + 4$, slope $m=\frac{3}{2}$

Step2: Use point-slope form

$y - 5 = \frac{3}{2}(x - 2)$

Step3: Simplify and convert to standard form

$2(y - 5) = 3(x - 2) \implies 2y - 10 = 3x - 6 \implies 3x - 2y = -4$
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Answer:

Part 1:

1. Parallel
2. Coinciding
3. Perpendicular
4. None of these
5. Parallel
6. Coinciding

Part 2:

1. $2x + y = 4$
2. $3x - y = -1$
3. $3x - 2y = -4$