Solved Exercises on Using Equations to Model Situations in Grade 8

Master using equations to model situations: linear equations, word problems, and real-world applications through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Phone Plan Model
Exercise 1
A phone company charges $25 per month plus $0.10 per minute for calls. Write an equation to model the monthly cost and find the cost for 150 minutes.
Definition:

Linear Equation Model: An equation of the form y = mx + b that represents a real-world situation with a constant rate of change

Equation modeling process:
  1. Identify the dependent and independent variables
  2. Identify the constant (y-intercept) and rate (slope)
  3. Write the equation in y = mx + b form
  4. Use the equation to solve specific problems
Variables
Cost = C, Minutes = m
Equation
C = 0.10m + 25
Solution
C(150) = $40
Step 1: Identify variables

Independent variable: minutes (m) - what we can change

Dependent variable: cost (C) - what depends on minutes

Step 2: Identify components

Fixed monthly charge: $25 (y-intercept)

Rate per minute: $0.10 (slope)

Step 3: Write the equation

Total cost = Fixed charge + (Rate × Minutes)

C = 0.10m + 25

Step 4: Solve for specific case

For 150 minutes: C = 0.10(150) + 25 = 15 + 25 = $40

C = 0.10m + 25, C(150) = $40
Final answer:

The equation is C = 0.10m + 25, and the cost for 150 minutes is $40

Applied rules:

• Linear form: y = mx + b where m is rate and b is initial value

• Variable identification: Determine which quantity depends on another

• Unit consistency: Ensure all units match in the equation

2 Balance Model
Exercise 2
Sarah has $50 in her savings account. She deposits $15 each week. Write an equation to model her balance after w weeks. After how many weeks will she have $125?
Definition:

Balance Model: An equation that tracks how an initial amount changes over time due to regular additions or subtractions

Initial Amount
$50
Weekly Deposit
$15
Equation
B = 15w + 50
Step 1: Identify variables

Independent: weeks (w)

Dependent: balance (B)

Step 2: Identify initial amount and rate

Starting balance: $50

Deposit rate: $15 per week

Step 3: Write the equation

B = 15w + 50

Step 4: Find when balance reaches $125

125 = 15w + 50

75 = 15w

w = 5 weeks

B = 15w + 50, w = 5 weeks
Final answer:

The equation is B = 15w + 50, and Sarah will have $125 after 5 weeks

Applied rules:

• Balance equation: Final amount = Initial + (rate × time)

• Solving for time: Isolate the variable to find when a target is reached

• Verification: Check that solution makes sense in context

3 Distance Model
Exercise 3
A car travels at a constant speed of 60 mph. Write an equation to model the distance traveled after t hours. How far does it travel in 3.5 hours?
Definition:

Distance Model: An equation using the formula distance = rate × time to model motion at constant speed

Rate
60 mph
Time
t hours
Equation
d = 60t
Step 1: Identify the formula

Distance = Rate × Time

Step 2: Identify variables

Distance (d) depends on time (t)

Rate = 60 mph (constant)

Step 3: Write the equation

d = 60t

Step 4: Calculate distance for 3.5 hours

d = 60 × 3.5 = 210 miles

d = 60t, d(3.5) = 210 miles
Final answer:

The equation is d = 60t, and the car travels 210 miles in 3.5 hours

Applied rules:

• Distance formula: d = rt (distance = rate × time)

• Constant rate: Creates linear relationship

• Unit conversion: Ensure consistent time units

Equation Modeling Fundamentals
y = mx + b
Linear Equation Form
Slope (m)
rate of change
How y changes per unit x
Y-intercept (b)
starting value
Value of y when x = 0
Independent Variable
input (x)
What you can control
Key definitions:

Equation Model: A mathematical equation that represents a real-world situation

Independent Variable: The variable that can be changed (input)

Dependent Variable: The variable that changes in response (output)

Linear Model: An equation of the form y = mx + b with constant rate of change

Rate of Change: How much the dependent variable changes per unit of independent variable

Initial Value: The value of the dependent variable when the independent variable is 0

Model Validation: Checking that the equation makes sense in the real-world context

Domain Restriction: Limiting the input values based on the real-world situation

Equation Modeling Process:
  1. Read the problem: Understand the real-world situation
  2. Identify variables: Determine independent and dependent variables
  3. Find rate and initial value: Identify the slope and y-intercept
  4. Write the equation: Put in y = mx + b form
  5. Validate the model: Check that it makes sense
  6. Solve problems: Use the equation to answer questions
Tip 1: Always define your variables clearly with units.
Tip 2: Look for words like "per," "each," or "rate" to identify the slope.
Tip 3: Find the starting amount for the y-intercept.
Tip 4: Always check if your answer makes sense in the real-world context.
Common errors: Misidentifying variables, confusing rate with initial value, not checking units.
Exam preparation: Practice identifying key words, writing equations, solving for specific values.
Solution: Exercises 4 to 5
4 Cost Model
Exercise 4
A store sells notebooks for $3.50 each. There's a shipping fee of $5.00 for any order. Write an equation for the total cost of buying n notebooks. How many notebooks can be bought for $33?
Definition:

Cost Model: An equation that calculates total cost including fixed fees and variable costs

Variable Cost
$3.50 per notebook
Fixed Cost
$5.00 shipping
Equation
C = 3.50n + 5
Step 1: Identify components

Variable cost: $3.50 per notebook

Fixed cost: $5.00 shipping fee

Step 2: Identify variables

Independent: number of notebooks (n)

Dependent: total cost (C)

Step 3: Write the equation

Total cost = (cost per notebook × number of notebooks) + shipping fee

C = 3.50n + 5

Step 4: Find number of notebooks for $33

33 = 3.50n + 5

28 = 3.50n

n = 8 notebooks

C = 3.50n + 5, n = 8 notebooks
Final answer:

The equation is C = 3.50n + 5, and 8 notebooks can be bought for $33

Applied rules:

• Total cost: Fixed + Variable costs

• Isolating variable: Subtract fixed amount first

• Verification: 3.50(8) + 5 = 28 + 5 = $33 ✓

5 Volume Model
Exercise 5
A tank initially contains 100 gallons of water. Water flows out at a rate of 4 gallons per minute. Write an equation for the amount of water remaining after t minutes. How long until the tank is half empty?
Definition:

Volume Model: An equation that tracks how a quantity changes over time due to inflow or outflow

Initial Volume
100 gallons
Flow Rate
-4 gal/min
Equation
V = -4t + 100
Step 1: Identify the situation

Water is flowing OUT of the tank, so volume decreases

Rate = -4 gallons per minute (negative because decreasing)

Step 2: Identify variables

Independent: time (t) in minutes

Dependent: volume (V) in gallons

Step 3: Write the equation

Volume = Initial volume + (rate × time)

V = 100 + (-4)t = 100 - 4t

Or: V = -4t + 100

Step 4: Find time when tank is half empty

Half empty means half full: V = 100/2 = 50 gallons

50 = -4t + 100

-50 = -4t

t = 12.5 minutes

V = -4t + 100, t = 12.5 min
Final answer:

The equation is V = -4t + 100, and the tank is half empty after 12.5 minutes

Applied rules:

• Decreasing rate: Negative slope indicates reduction

• Half empty: Half of initial volume

• Domain restriction: Time must be between 0 and 25 minutes (when tank is empty)

Equation Modeling Analysis Summary
y = mx + b
Linear Model Standard Form
Key definitions:

Equation Model: A mathematical representation of a real-world situation using an equation

Independent Variable: The variable that can be controlled or manipulated (x-axis)

Dependent Variable: The variable that changes in response (y-axis)

Linear Model: An equation of the form y = mx + b representing constant rate of change

Slope (m): The rate of change, representing how much y changes per unit of x

Y-intercept (b): The initial value when x = 0

Model Validation: Checking that the equation accurately represents the real-world situation

Domain Restriction: Limiting input values based on the real-world context

Complete Equation Modeling Process:
  1. Problem comprehension: Read and understand the real-world situation
  2. Variable identification: Determine which quantities depend on others
  3. Component analysis: Identify the rate of change and initial value
  4. Equation formation: Write in y = mx + b form
  5. Model validation: Check that the equation makes sense in context
  6. Problem solving: Use the equation to answer specific questions
  7. Result interpretation: Translate mathematical results back to real-world meaning
Tip 1: Look for keywords like "per," "each," "rate," or "for every" to identify the slope.
Tip 2: The y-intercept is often the "starting" or "initial" value mentioned in the problem.
Tip 3: Negative slopes indicate decreasing quantities over time.
Tip 4: Always verify your solution by substituting back into the original equation.
Applications: Used in finance, physics, business, engineering, and everyday life decisions.
Limitations: Linear models assume constant rate of change, which may not always be realistic.
Essential Rules:

• Linear form: y = mx + b, where m is slope and b is y-intercept

• Rate identification: Slope comes from "per" or "rate" in problem

• Initial value: Y-intercept is value when independent variable is 0

• Unit consistency: Ensure all variables use compatible units

• Context validation: Solutions must make sense in real-world situation

Questions & Answers

Question: How do I know which variable should be the independent variable and which should be dependent?

Answer: The independent variable is what you can control or change, while the dependent variable is what responds to that change:

  • Independent (x): The "input" or "cause" - what you manipulate
  • Dependent (y): The "output" or "effect" - what you measure

Ask yourself: "What am I changing to see what happens?" The thing you change is independent; what happens is dependent.

In a phone plan, minutes used (x) affects cost (y), so minutes are independent and cost is dependent.

Question: What if the rate of change isn't constant? Can I still use a linear equation?

Answer: No, linear equations only work when the rate of change is constant:

  • Linear: Constant rate of change (straight line graph)
  • Non-linear: Changing rate of change (curved graph)

Examples of non-linear situations:
- Population growth (accelerating)
- Area of a circle as radius increases (quadratic)
- Compound interest (exponential)

For grade 8, focus on linear equations where the rate stays the same throughout the situation.

Question: How do I handle negative slopes in real-world problems? What do they mean?

Answer: Negative slopes indicate that as the independent variable increases, the dependent variable decreases:

Common real-world examples:
- Water draining from a tank (volume decreases over time)
- Money spent from a bank account (balance decreases with spending)
- Temperature cooling down (temperature decreases over time)

The slope value tells you the rate of decrease. For example, if V = -4t + 100, the volume decreases by 4 gallons per minute.

Always consider what a negative slope means in the context of your problem.

Question: What if my equation gives a negative answer when the context requires a positive value?

Answer: This indicates a domain restriction - the equation is only valid for certain input values:

For example, in the water tank problem V = -4t + 100:
- The tank starts with 100 gallons at t = 0
- The tank empties at t = 25 (when V = 0)
- The equation is meaningless for t > 25 (negative volume)

Always consider the realistic domain for your variables. In real-world problems, the equation only applies within certain limits.

If you get a negative result that doesn't make sense, re-examine the domain restrictions.

Question: How do I check if my equation model is correct?

Answer: Use multiple validation methods:

  1. Check initial conditions: Does your equation match the starting situation?
  2. Test with known values: Use values from the problem to verify
  3. Verify rate: Does the slope match the stated rate of change?
  4. Check reasonableness: Does the answer make sense in context?
  5. Substitute back: Plug solutions back into the original equation

For example, if you wrote C = 0.10m + 25 for the phone plan, verify:
- When m = 0, C = 25 (initial monthly fee)
- Each additional minute increases cost by $0.10
- For m = 150, C = $40 (matches your calculation)