Writing Linear Equations
Learn to write linear equations from given information like slope, points, or real-world situations.
Definition
- The slope () - how steep the line is
- A point on the line - or the y-intercept ()
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Worked Examples
Write the equation of a line with slope and y-intercept .
Identify the slope
→ Slope is 3
Identify the y-intercept
→ Y-intercept is
Substitute into slope-intercept form
→
Answer:
Common Mistakes
Confusing slope and y-intercept positions
Why it's wrong: In , the coefficient of is ALWAYS the slope, and the constant is ALWAYS the y-intercept.
Correct: Remember: multiplies (slope), stands alone (y-intercept). In , the slope is 3, not 5!
Calculating slope as
Why it's wrong: This gives the reciprocal of the correct slope!
Correct: Slope is always (y's on top, x's on bottom)
Forgetting to distribute negative slopes
Why it's wrong: When and the point is : becomes , not
Correct: A negative times a negative is positive:
Using inconsistent points in slope formula
Why it's wrong: Switching the order of points mid-calculation gives wrong answers.
Correct: If you start with , use that consistently:
Interactive Visual
Linear Function Explorer
| x | y |
|---|---|
| -2 | -2 |
| -1 | -1 |
| 0 | 0 |
| 1 | 1 |
| 2 | 2 |
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y = 2x + 1
m=2, b=1
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Practice Problems
16 problemsWhat is the equation of a line with slope and y-intercept ?
Why It Matters
- Business: A company charges 50 dollars per hour plus a 100 dollar service fee:
- Science: Temperature drops 2 degrees every hour from an initial 20 degrees:
- Finance: You save 25 dollars per week starting with 150 dollars:
- Sports: A runner covers 8 meters per second:
Real World Applications
Cell Phone Plans
Phone companies use linear equations to calculate monthly bills with base fees and per-unit charges.
Example:
A plan costs 30 dollars per month plus 0.10 dollars per text. The equation is .
A streaming service charges 12 dollars per month plus 3 dollars for each premium movie rental.
Write an equation for the monthly cost based on premium rentals.
Step 1: Write the mathematical expression
Let be the number of rentals. Write the cost equation:
Science Experiments
Scientists use linear equations to model relationships like temperature changes or distance traveled.
Example:
Water cools by 5 degrees per minute from an initial 80 degrees:
A candle loses 2 centimeters of height per hour. It starts at 20 cm tall.
Write an equation for the candle's height after hours.
Step 1: Write the mathematical expression
Height equation:
Savings Goals
Linear equations help track savings progress toward financial goals.
Example:
Starting with 200 dollars and saving 50 dollars weekly:
You have 75 euros saved and add 15 euros each week from your allowance.
Write an equation for your total savings after weeks.
Step 1: Write the mathematical expression
Savings equation:
Key Takeaways
- 1Use when you know the slope and y-intercept directly
- 2Use point-slope form when you know a point and slope
- 3Calculate slope from two points:
- 4From tables: slope is the constant change in divided by change in
- 5In word problems: the rate is the slope, the starting value is the y-intercept
Frequently Asked Questions
Glossary
- Slope-intercept form
- The equation where is the slope and is the y-intercept
- Point-slope form
- The equation using slope and point
- Y-intercept
- The point where the line crosses the y-axis; the value of when
- Rate of change
- Another name for slope; how much changes for each unit change in
Formula Card
Slope-Intercept Form
Use when you know the slope and y-intercept
Point-Slope Form
Use when you know a point and the slope
Slope Formula
Calculate slope from two points