Linear relationship word problems: foundations

[Video Placeholder - Introduction to linear relationships]

Duration: 4:45

Note: Many real-world situations can be modeled with linear relationships. On the SAT, you'll need to interpret these relationships and solve problems using them.

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Understanding linear relationships

A linear relationship can be represented by the equation:

y = mx + b

Where:

m is the slope (rate of change)
b is the y-intercept (initial value)
x is the independent variable
y is the dependent variable

[Graph showing a linear relationship with positive slope]

Key features of linear relationships

Constant rate of change
Straight-line graph
Can be increasing or decreasing
Proportional when b = 0

Example 1: Rate problem

A car is traveling at a constant speed of 60 miles per hour. Write an equation that relates the distance traveled (d) to the time spent driving (t). How far will the car travel in 3.5 hours?

Step 1: Identify variables and relationship

Distance = rate × time

d = 60 × t

Step 2: Substitute the given time

d = 60 × 3.5

d = 210

Answer: The equation is d = 60t, and the car will travel 210 miles in 3.5 hours.

Example 2: Initial value problem

A swimming pool has 200 gallons of water in it. A hose is adding water at a rate of 15 gallons per minute. Write an equation for the total amount of water (W) in the pool after m minutes. How much water will be in the pool after 30 minutes?

Step 1: Identify initial value and rate

Initial amount (b) = 200 gallons

Rate (m) = 15 gallons per minute

Step 2: Write the equation

W = 15m + 200

Step 3: Substitute m = 30

W = 15(30) + 200

W = 450 + 200 = 650

Answer: The equation is W = 15m + 200, and there will be 650 gallons after 30 minutes.

Try it!

A phone plan charges a $20 monthly fee plus $0.10 per text message. Write an equation for the total monthly cost (C) based on the number of text messages (t). What would be the cost for 150 text messages?

Step 1: Identify the components

Fixed cost: $ (monthly fee)

Variable cost: $ per text message

Step 2: Write the equation

C = t +

Step 3: Calculate cost for 150 texts

C = $

The total cost has two parts: a fixed monthly fee and a variable cost that depends on the number of texts.

Interpreting linear relationships

Example 3: Real-world interpretation

The equation C = 4.5p + 25 represents the cost (C) of printing p posters. What do the slope and y-intercept represent in this context?

Slope interpretation:

The slope is 4.5 (coefficient of p)

This represents the cost per poster ($4.50 each)

Y-intercept interpretation:

The y-intercept is 25 (constant term)

This represents the fixed setup cost ($25) regardless of how many posters are printed

Interpretation practice

The equation T = 0.07s + 50 represents the total pay (T) a salesperson earns based on their sales (s). What does the slope represent?

What does the y-intercept represent?

Up next: Video - Comparing linear relationships