Linear Function Word Problems | Worksheet - Free Printable

Problem Analysis:

The task involves solving linear function word problems by expressing relationships in the slope-intercept form \( y = mx + b \), where:
- \( m \) is the slope (rate of change),
- \( b \) is the y-intercept (initial value when \( x = 0 \)).

We will solve each problem step by step.

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Problem 1: Cell Phone Plan

Description:
Chandler's cell phone plan costs $20 plus $5 per gigabyte of data he uses. Write an equation in slope-intercept form to represent this linear function. If Chandler uses 3 gigabytes of data this month, how much will his cell phone bill be?

#### Step 1: Identify the components
- Slope (\( m \)): Cost per gigabyte of data = $5.
- Y-intercept (\( b \)): Fixed monthly cost = $20.
- Variable (\( x \)): Gigabytes of data used.
- Output (\( y \)): Total cell phone bill.

#### Step 2: Write the equation
The general form is \( y = mx + b \). Substituting the values:
\[ y = 5x + 20 \]

#### Step 3: Calculate the bill for 3 gigabytes
Substitute \( x = 3 \) into the equation:
\[ y = 5(3) + 20 \]
\[ y = 15 + 20 \]
\[ y = 35 \]

#### Final Answer for Problem 1:
\[ \boxed{y = 5x + 20} \]
If Chandler uses 3 gigabytes, his bill will be:
\[ \boxed{35} \]

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Problem 2: Gym Membership

Description:
A membership at Rocky Heights Climbing Gym costs $36 per month plus a $40 initial fee. Write an equation in slope-intercept form to represent this linear function. If Meg gets a membership that lasts 6 months, how much does she pay?

#### Step 1: Identify the components
- Slope (\( m \)): Monthly cost = $36.
- Y-intercept (\( b \)): Initial fee = $40.
- Variable (\( x \)): Number of months.
- Output (\( y \)): Total cost.

#### Step 2: Write the equation
The general form is \( y = mx + b \). Substituting the values:
\[ y = 36x + 40 \]

#### Step 3: Calculate the total cost for 6 months
Substitute \( x = 6 \) into the equation:
\[ y = 36(6) + 40 \]
\[ y = 216 + 40 \]
\[ y = 256 \]

#### Final Answer for Problem 2:
\[ \boxed{y = 36x + 40} \]
If Meg gets a membership for 6 months, she pays:
\[ \boxed{256} \]

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Problem 3: Soccer Practice

Description:
Luna's soccer team always starts practice with a 12-minute warm-up. Then, the team spends 8 minutes on drills for every goal they scored in their last game. Write an equation in slope-intercept form to represent this linear function. If tomorrow's practice is 60 minutes long, how many drills can Luna's team practice?

#### Step 1: Identify the components
- Slope (\( m \)): Minutes spent on drills per goal = 8.
- Y-intercept (\( b \)): Warm-up time = 12 minutes.
- Variable (\( x \)): Number of goals scored.
- Output (\( y \)): Total practice time.

#### Step 2: Write the equation
The general form is \( y = mx + b \). Substituting the values:
\[ y = 8x + 12 \]

#### Step 3: Solve for the number of drills when \( y = 60 \)
Substitute \( y = 60 \) into the equation:
\[ 60 = 8x + 12 \]
Subtract 12 from both sides:
\[ 48 = 8x \]
Divide by 8:
\[ x = 6 \]

#### Final Answer for Problem 3:
\[ \boxed{y = 8x + 12} \]
If tomorrow's practice is 60 minutes long, Luna's team can practice:
\[ \boxed{6} \] drills.

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Problem 4: Tablet Savings

Description:
Vivian is saving up to buy a tablet. She already has $100 saved, and she plans to save $20 per week. Write an equation in slope-intercept form to represent this linear function. If the tablet Vivian wants to buy is $520, how many weeks will it take her to save enough for the tablet?

#### Step 1: Identify the components
- Slope (\( m \)): Weekly savings = $20.
- Y-intercept (\( b \)): Initial savings = $100.
- Variable (\( x \)): Number of weeks.
- Output (\( y \)): Total savings.

#### Step 2: Write the equation
The general form is \( y = mx + b \). Substituting the values:
\[ y = 20x + 100 \]

#### Step 3: Solve for the number of weeks when \( y = 520 \)
Substitute \( y = 520 \) into the equation:
\[ 520 = 20x + 100 \]
Subtract 100 from both sides:
\[ 420 = 20x \]
Divide by 20:
\[ x = 21 \]

#### Final Answer for Problem 4:
\[ \boxed{y = 20x + 100} \]
It will take Vivian:
\[ \boxed{21} \] weeks to save enough for the tablet.

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Summary of Final Answers:

1. \( \boxed{y = 5x + 20} \) and \( \boxed{35} \)
2. \( \boxed{y = 36x + 40} \) and \( \boxed{256} \)
3. \( \boxed{y = 8x + 12} \) and \( \boxed{6} \)
4. \( \boxed{y = 20x + 100} \) and \( \boxed{21} \)