Linear Functions Homework Worksheet: Writing Linear Equations and Applications

Let’s work through each problem one by one.

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Problem 1: slope = -6; passes through (-4, 1)

We need to write the equation in slope-intercept form:
> y = mx + b
where m is the slope, and b is the y-intercept.

We know:
- m = -6
- Point (-4, 1) → x = -4, y = 1

Plug into the equation to find b:

1 = (-6)(-4) + b
1 = 24 + b
b = 1 - 24 = -23

✔ So the equation is:
y = -6x - 23

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Problem 2: slope = 4/3; passes through (-5, -6)

Again, use y = mx + b

m = 4/3
Point: x = -5, y = -6

-6 = (4/3)(-5) + b
-6 = -20/3 + b

Convert -6 to thirds: -6 = -18/3

So:
-18/3 = -20/3 + b
Add 20/3 to both sides:
b = 2/3

✔ Equation:
y = (4/3)x + 2/3

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Problem 3: passes through (-4, 11) and (2, 8)

First, find the slope:

m = (y₂ - y₁)/(x₂ - x₁) = (8 - 11)/(2 - (-4)) = (-3)/(6) = -1/2

Now use point-slope to find b. Let’s use point (2, 8):

y = mx + b
8 = (-1/2)(2) + b
8 = -1 + b
b = 9

✔ Equation:
y = (-1/2)x + 9

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Problem 4: passes through (6, 3) and (14, -5)

Find slope:

m = (-5 - 3)/(14 - 6) = (-8)/8 = -1

Use point (6, 3):

3 = (-1)(6) + b
3 = -6 + b
b = 9

✔ Equation:
y = -x + 9

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Problem 5: Troy’s truck

Gas tank = 30 gallons
Miles per gallon = 21

a) Let:
- x = number of miles driven
- y = gallons of gas left

He starts with 30 gallons. For every mile, he uses 1/21 gallon.

So:
y = 30 - (1/21)x

✔ Equation:
y = 30 - (1/21)x

b) If tank is half full → y = 15

15 = 30 - (1/21)x
Subtract 30:
-15 = -(1/21)x
Multiply both sides by -21:
x = 315

✔ Troy drove 315 miles

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Problem 6: Mikayla’s training

Starts at 2 miles
Increases by 3/4 mile each week

a) Let:
- x = number of weeks since start
- y = distance she can run (in miles)

Each week adds 0.75 miles.

So:
y = 2 + (3/4)x

✔ Equation:
y = 2 + (3/4)x

b) Half-marathon = 13.1 miles
After 12 weeks → x = 12

y = 2 + (3/4)(12) = 2 + 9 = 11 miles

11 < 13.1 → Not ready

✔ She will not be ready after 12 weeks.

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Final Answer:
1. y = -6x - 23
2. y = (4/3)x + 2/3
3. y = (-1/2)x + 9
4. y = -x + 9
5a. y = 30 - (1/21)x
5b. 315 miles
6a. y = 2 + (3/4)x
6b. No, she will not be ready.