Slope-intercept form: write an equation from a word problem

Key notes

Slope-Intercept Form Recap

The slope-intercept form of a line is:

y=mx+b

where:

• m = slope (rate of change)
• b = y-intercept (starting value)

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Word Problems and What They Mean

In real-life problems:

• Slope (m) often represents the rate of change (how much y changes for each unit of x).
• Y-intercept (b) often represents the initial value (when x=0).
• x is usually the independent variable (time, distance, items, etc.).
• y is usually the dependent variable (total cost, height, money earned, etc.).

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Steps to Write an Equation from a Word Problem

• Identify the slope m from the rate given in the problem.
• Identify the y-intercept b(starting amount or initial value).
• Write the equation in:

y=mx+b

• Label the variables to match the problem’s context.

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Example 1 – Direct Information

A taxi charges a base fare of $5 plus $2 for every mile.

• Base fare = b=5 (starting cost)
• Rate = m=2 (cost per mile)

Equation:

y=2x+5

Where:

• x = miles driven
• y= total cost in dollars

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Example 2 – Given Two Situations

A gym charges a joining fee plus a monthly fee. You pay $50 after 1 month and $80 after 4 months.
Step 1: Find slope mmm
Points: (1, 50) and (4, 80)

m=80−50/4−1=30/3=10

Step 2: Find bbb using (1, 50)

50=10(1)+b

b=40

Step 3: Write equation

y=10x+40

Where:

• x = number of months
• y = total cost in dollars

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Special Tips

• “Starting at” or “initially” usually gives b.
• “Per”, “each”, “every” usually gives m.
• If given two data points, find mmm first, then b.
• Always define what x and y represent.

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Practice Problems

1. A streaming service charges $8 per month and a one-time sign-up fee of $12. Write the equation for the total cost y after x months.
2. A plumber charges $60 for the first hour and $40 for each additional hour. Write the equation for the total cost after x hours.
3. You have $100 in your savings account and deposit $15 each week. Write the equation for the total savings after x weeks.
4. A phone company charges $25 per month and a $50 activation fee. Write the cost equation.

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Learn with an example