Question

skill #7: linear equations

1. identify the slope and the y-intercept of the following linear equations.

a. $y = \frac{1}{3}x + 9$ b. $y = -2x$
c. $2x - 4y = 12$ d. $3x + 4y = 18$

1. find the equation in slope-intercept form of a line with a slope of -2 and a y-intercept of -9.

1. find the equation of a line in slope-intercept form that has a slope of 4 and passes through (3, 8).

1. find the equation of a line in slope-intercept form that passes through (2, 5) and (-2, 3).

graphing calculator: https://www.desmos.com/calculator (available in app store)
scientific calculator: https://www.desmos.com/scientific (available in app store)

Explanation:

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Problem 44: Identify slope and y-intercept

Slope-intercept form is $y=mx+b$, where $m$ = slope, $b$ = y-intercept.

Part a

Step1: Match to slope-intercept form

Equation: $y=\frac{1}{3}x+9$
Compare to $y=mx+b$: $m=\frac{1}{3}$, $b=9$

Part b

Step1: Rewrite to match slope-intercept form

Equation: $y=-2x = -2x+0$
Compare to $y=mx+b$: $m=-2$, $b=0$

Part c

Step1: Isolate $y$ to convert form

$2x-4y=12$
Subtract $2x$: $-4y=-2x+12$

Step2: Solve for $y$

Divide by $-4$: $y=\frac{-2}{-4}x+\frac{12}{-4}$
Simplify: $y=\frac{1}{2}x-3$
Compare to $y=mx+b$: $m=\frac{1}{2}$, $b=-3$

Part d

Step1: Isolate $y$ to convert form

$3x+4y=18$
Subtract $3x$: $4y=-3x+18$

Step2: Solve for $y$

Divide by $4$: $y=\frac{-3}{4}x+\frac{18}{4}$
Simplify: $y=-\frac{3}{4}x+\frac{9}{2}$
Compare to $y=mx+b$: $m=-\frac{3}{4}$, $b=\frac{9}{2}$

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Problem 45: Write slope-intercept equation

Step1: Substitute $m$ and $b$ into form

Given $m=-2$, $b=-9$
Slope-intercept form: $y=mx+b$
Substitute values: $y=-2x+(-9)$
Simplify: $y=-2x-9$

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Problem 46: Write equation from slope and point

Step1: Use point-slope form

Point-slope formula: $y-y_1=m(x-x_1)$
Given $m=4$, $(x_1,y_1)=(3,8)$
Substitute: $y-8=4(x-3)$

Step2: Convert to slope-intercept form

Expand right side: $y-8=4x-12$
Add 8 to both sides: $y=4x-12+8$
Simplify: $y=4x-4$

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Problem 47: Write equation from two points

Step1: Calculate slope $m$

Slope formula: $m=\frac{y_2-y_1}{x_2-x_1}$
Given $(x_1,y_1)=(2,5)$, $(x_2,y_2)=(-2,3)$
Substitute: $m=\frac{3-5}{-2-2}=\frac{-2}{-4}=\frac{1}{2}$

Step2: Use point-slope form

Substitute $m=\frac{1}{2}$ and $(2,5)$: $y-5=\frac{1}{2}(x-2)$

Step3: Convert to slope-intercept form

Expand right side: $y-5=\frac{1}{2}x-1$
Add 5 to both sides: $y=\frac{1}{2}x-1+5$
Simplify: $y=\frac{1}{2}x+4$

Answer:

44.

a. Slope: $\frac{1}{3}$, y-intercept: $9$
b. Slope: $-2$, y-intercept: $0$
c. Slope: $\frac{1}{2}$, y-intercept: $-3$
d. Slope: $-\frac{3}{4}$, y-intercept: $\frac{9}{2}$

45.

$y=-2x-9$

46.

$y=4x-4$

47.

$y=\frac{1}{2}x+4$