Question

1. write the equation in slope-intercept form of the line that is parallel to the graph of each equation and passes through the given point.

a) $y = 3x + 6, \quad (4,7)$
b) $y = x - 4, \quad (-2,3)$

1. find the slope of the following points.

a) $(19, -16) \quad (-7, -15)$
b) $(9,16) \quad (20,8)$

Explanation:

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Problem 3a:

Step1: Identify parallel slope

Parallel lines have equal slopes. For $y=3x+6$, slope $m=3$.

Step2: Use point-slope form

Point-slope formula: $y-y_1=m(x-x_1)$. Substitute $m=3$, $(x_1,y_1)=(4,7)$:
$y-7=3(x-4)$

Step3: Convert to slope-intercept

Simplify to $y=mx+b$:
$y-7=3x-12$
$y=3x-12+7$
$y=3x-5$

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Problem 3b:

Step1: Identify parallel slope

For $y=x-4$, slope $m=1$.

Step2: Use point-slope form

Substitute $m=1$, $(x_1,y_1)=(-2,3)$:
$y-3=1(x-(-2))$

Step3: Convert to slope-intercept

Simplify to $y=mx+b$:
$y-3=x+2$
$y=x+2+3$
$y=x+5$

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Problem 4a:

Step1: Use slope formula

Slope formula: $m=\frac{y_2-y_1}{x_2-x_1}$. Substitute $(x_1,y_1)=(19,-16)$, $(x_2,y_2)=(-7,-15)$:
$m=\frac{-15-(-16)}{-7-19}$

Step2: Calculate the value

$m=\frac{-15+16}{-26}=\frac{1}{-26}=\frac{-1}{26}$

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Problem 4b:

Step1: Use slope formula

Substitute $(x_1,y_1)=(9,16)$, $(x_2,y_2)=(20,8)$:
$m=\frac{8-16}{20-9}$

Step2: Calculate the value

$m=\frac{-8}{11}$

Answer:

a) $y=3x-5$
b) $y=x+5$
a) $\frac{-1}{26}$
b) $\frac{-8}{11}$