How do I know which number is m and which is b?

The starting amount—the value when you haven't used the service or zero time has passed—is always b. The rate of change—the amount per unit (per hour, per class, per item)—is always m. Look for words like 'per,' 'each,' or 'every' to spot the rate.

What if the problem says the quantity is decreasing? Do I make m negative?

Yes. If the problem uses words like 'loses,' 'burns,' 'decreases,' 'shrinks,' or 'drains,' the quantity is going down, so m is negative. For example, a candle burning means m is negative because height decreases over time.

Do I have to use x and y as my variables?

No. You can use any letters that make sense. In a cost problem, you might use C for cost and t for time. In a height problem, you might use h for height and m for minutes. Always define your variables clearly at the start so your equation is easy to understand.

What does y = mx + b actually mean in a real situation?

It means: 'The final quantity (y) equals (the rate times the number of times the rate happens) plus (the starting amount).' For a gym with a 20-dollar fee and 5 dollars per class, y = 5x + 20 means 'total cost equals 5 dollars per class times the number of classes, plus the 20-dollar sign-up fee.'

M8MATH-7.2

Constructing Linear Models from Descriptions

Learn to translate real-world situations into linear equations y = mx + b by identifying the starting amount and constant rate.

What you'll do in this lesson

A voice-first session with the Crimsora tutor on Constructing Linear Models from Descriptions, then targeted practice and FRQs — with the tutor adapting to where you get stuck.

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What this lesson covers

Every time you sign up for an online service, join a gym, or watch a candle burn down, you're working with linear relationships. A fitness app might charge a 20-dollar sign-up fee plus 5 dollars per class. A candle that starts at 12 cm tall and shrinks 0.5 cm every hour follows a pattern too. In this lesson, you'll learn to translate these everyday situations into linear equations in the form

y = mx + b

. This skill lets you write a formula that predicts costs, heights, distances, or any quantity that changes at a steady rate.

Understanding Starting Amount and Rate

A linear model has two key ingredients: a starting amount and a constant rate of change. The starting amount is the initial value—what you have or what something costs before any change happens. When you join a gym for 20 dollars with a sign-up fee, that 20 dollars is your starting amount even if you haven't attended a single class yet. The constant rate of change is how much the quantity increases or decreases per unit. In the gym example, you pay 5 dollars per class, so the rate is 5 dollars per class.

The starting amount becomes the

b

value (the y-intercept), and the rate of change becomes the

m

value (the slope) in

y = mx + b

. Notice that the rate is always expressed per unit—per class, per hour, per day, per item purchased. If a problem says 'plus 5 dollars per class,' that 5 is your rate. If it says 'minus 2 cm per hour' (a candle shrinking), the rate is negative:

m = -2

Identifying m and b from Words

To build a linear equation from a description, follow these steps: First, define your variables clearly. Choose what

x

represents (usually the quantity that changes independently—time, number of items, number of classes) and what

y

represents (usually the quantity you're measuring—total cost, height, distance). Next, find the starting amount—the value of

y

when

x = 0

. This is your

b

value. Then, find the constant rate of change. Look for language like 'per,' 'each,' 'every,' or 'for each.' This is your

m

value. Remember: if the quantity is decreasing,

m

is negative.

Example: 'A candle is 12 cm tall and burns 0.5 cm per hour.' Let

x =

hours and

y =

height in cm. The starting amount is 12 cm (when

x = 0

), so

b = 12

. The candle burns 0.5 cm per hour, meaning it decreases, so

m = -0.5

. The equation is

y = -0.5x + 12

Writing Equations from Real Situations

Once you can identify

m

and

b

, writing the equation is straightforward: substitute into

y = mx + b

. Let's work through a fitness app example: 'Sign-up fee is 20 dollars, and each class costs 5 dollars.' Define:

x =

number of classes,

y =

total cost in dollars. The sign-up fee of 20 dollars is the starting amount (when you've attended zero classes), so

b = 20

. Each class costs 5 dollars, so

m = 5

. The equation is

y = 5x + 20

.

Another example with a negative rate: 'A pool is being drained. It starts with 5,000 liters and loses 50 liters per minute.' Define:

x =

minutes,

y =

liters remaining. Starting amount: 5,000 liters, so

b = 5000

. The pool is losing water, so the rate is negative:

m = -50

. The equation is

y = -50x + 5000

. These equations let you predict the cost at any number of classes or the water volume at any time.

Common Mistakes and How to Avoid Them

Students often mix up which number is

m

and which is

b

. A useful check: the starting amount (

b

) is always what happens at time zero or before any change occurs. The rate (

m

) is always paired with words like 'per' or 'each.' Confusing these leads to equations that don't match the situation.

Another common error is forgetting to make the rate negative when a quantity is decreasing. If a problem says 'loses,' 'burns,' 'shrinks,' or 'decreases,' the rate is negative. Writing

y = 50x + 5000

for a draining pool is wrong; it would predict the pool growing, not shrinking.

Also, be careful with units and definitions. Always write down what

x

and

y

represent before you write the equation. This prevents mixing up rate and starting amount, and it makes it easy to check whether your answer makes sense.

Key terms

Linear model.: An equation of the form $y = mx + b$ that represents a relationship between two quantities that change at a constant rate.
Initial value (or starting amount).: The value of $y$ when $x = 0$ ; represented by $b$ in $y = mx + b$ .
Rate of change (or constant rate).: The amount $y$ changes for each unit increase in $x$ ; represented by $m$ in $y = mx + b$ .
Slope ( $m$ ).: The constant rate of change in a linear equation; tells you how much $y$ increases or decreases per unit of $x$ .
y-intercept ( $b$ ).: The value of $y$ when $x = 0$ ; the starting amount in a linear model.
Variable.: A letter (usually $x$ or $y$ ) that represents an unknown or changing quantity in an equation.
Negative rate.: A rate of change less than zero, which occurs when a quantity is decreasing; results in a negative slope ( $m < 0$ ).

Worked example

A parking garage charges a 3-dollar entry fee and then 2 dollars for every hour you park. Write a linear equation that relates the total cost (in dollars) to the number of hours parked. Then use it to find the cost of parking for 5 hours.

Step 1: Define your variables.

Let

x =

number of hours parked, and let

y =

total cost in dollars.

Step 2: Identify the starting amount ( $b$ ).

The entry fee is paid immediately, before you park at all. This is the starting amount:

b = 3

dollars.

Step 3: Identify the rate of change ( $m$ ).

You pay 2 dollars for every hour parked. This is a constant rate being added, so

m = 2

dollars per hour.

Step 4: Write the equation.

Substitute

m = 2

and

b = 3

into

y = mx + b

y = 2x + 3

Step 5: Use the equation to find the cost of 5 hours of parking.

Substitute

x = 5

into the equation:

y = 2(5) + 3 = 10 + 3 = 13

The cost of parking for 5 hours is 13 dollars.

Practice questions

A mobile game has a 12-dollar upfront cost to download, and then you earn rewards worth 3 dollars every day you play. Write a linear equation where

y

is the total value of your account (in dollars) and

x

is the number of days played.

Answer: $y = 3x + 12$

The upfront cost of 12 dollars is the starting amount when

x = 0

(zero days played), so

b = 12

. Each day you earn 3 dollars, which is the constant rate of change, so

m = 3

. Substituting into

y = mx + b

gives

y = 3x + 12

An ice cube at room temperature starts 24 mm tall and melts 2 mm per minute. Which equation models the height of the ice cube (in mm) as a function of time (in minutes)? (A)

y = 2x + 24

(B)

y = -2x + 24

(C)

y = 24x - 2

(D)

y = 24x + 2

$y = 2x + 24$
$y = -2x + 24$
$y = 24x - 2$
$y = 24x + 2$

Answer: $y = -2x + 24$

Let

x =

minutes and

y =

height in mm. The ice cube starts at 24 mm, so

b = 24

. The ice melts (decreases) 2 mm per minute, so the rate is negative:

m = -2

. The equation is

y = -2x + 24

, which is choice (B). Choice (A) forgets the negative sign. Choices (C) and (D) confuse which variable is multiplied by which number.

A restaurant supply order costs 45 dollars for a bulk delivery fee plus 8 dollars for each box of supplies ordered. Write an equation for the total cost

C

in terms of the number of boxes

b

ordered. How much does an order of 12 boxes cost?

Answer: $C = 8b + 45$ ; the cost is 141 dollars.

Let

b =

number of boxes and

C =

total cost in dollars. The delivery fee of 45 dollars is the starting amount (you pay it even if

b = 0

), so the constant term is 45. Each box costs 8 dollars, so the coefficient of

b

is 8. The equation is

C = 8b + 45

. For 12 boxes:

C = 8(12) + 45 = 96 + 45 = 141

dollars.

FAQ

How do I know which number is $m$ and which is $b$ ?: The starting amount—the value when you haven't used the service or zero time has passed—is always $b$ . The rate of change—the amount per unit (per hour, per class, per item)—is always $m$ . Look for words like 'per,' 'each,' or 'every' to spot the rate.
What if the problem says the quantity is decreasing? Do I make $m$ negative?: Yes. If the problem uses words like 'loses,' 'burns,' 'decreases,' 'shrinks,' or 'drains,' the quantity is going down, so $m$ is negative. For example, a candle burning means $m$ is negative because height decreases over time.
Do I have to use $x$ and $y$ as my variables?: No. You can use any letters that make sense. In a cost problem, you might use $C$ for cost and $t$ for time. In a height problem, you might use $h$ for height and $m$ for minutes. Always define your variables clearly at the start so your equation is easy to understand.
What does $y = mx + b$ actually mean in a real situation?: It means: 'The final quantity (y) equals (the rate times the number of times the rate happens) plus (the starting amount).' For a gym with a 20-dollar fee and 5 dollars per class, $y = 5x + 20$ means 'total cost equals 5 dollars per class times the number of classes, plus the 20-dollar sign-up fee.'

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