Writing Linear Equations

Learn to write linear equations from given information like slope, points, or real-world situations.

Intermediate25 minLesson

Definition

Writing linear equations means creating an equation in the form

y = mx + b

(or equivalent) from given information.

To write a linear equation, you need:

The slope ( $m$ ) - how steep the line is
A point on the line - or the y-intercept ( $b$ )

\text{Given: slope and y-intercept} \rightarrow y = mx + b

\text{Given: slope and a point} \rightarrow y - y_1 = m(x - x_1)

Once you identify the slope and a point, you can write the equation!

Try it now

What is the equation of a line with slope

m = 2

and y-intercept

b = 5

Worked Examples

Write the equation of a line with slope $m = 3$ and y-intercept $b = -2$ .

Identify the slope

$m = 3$ → Slope is 3

Identify the y-intercept

$b = -2$ → Y-intercept is $-2$

Substitute into slope-intercept form

$y = mx + b$ $y = 3x + (-2)$ → $y = 3x - 2$

Answer: $y = 3x - 2$

Common Mistakes

Confusing slope and y-intercept positions

Why it's wrong: In $y = mx + b$ , the coefficient of $x$ is ALWAYS the slope, and the constant is ALWAYS the y-intercept.

Correct: Remember: $m$ multiplies $x$ (slope), $b$ stands alone (y-intercept). In $y = 5 + 3x$ , the slope is 3, not 5!

Calculating slope as $\frac{x_2 - x_1}{y_2 - y_1}$

Why it's wrong: This gives the reciprocal of the correct slope!

Correct: Slope is always $\frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1}$ (y's on top, x's on bottom)

Forgetting to distribute negative slopes

Why it's wrong: When $m = -2$ and the point is $(3, 5)$ : $y - 5 = -2(x - 3)$ becomes $y - 5 = -2x + 6$ , not $-2x - 6$

Correct: A negative times a negative is positive: $-2 \times (-3) = +6$

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 $(x_1, y_1) = (2, 5)$ , use that consistently: $m = \frac{y_2 - 5}{x_2 - 2}$

Interactive Visual

Linear Function Explorer

y = x

Slope (m)1

Y-Intercept (b)0

run

rise

x	y
-2	-2
-1	-1
0	0
1	1
2	2

Interactive Sandbox

Interactive Grapher

Try these examples:

y = 2x + 1

m=2, b=1

Expression Calculator

Try these:

History

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Practice Problems

16 problems

Problem 1 of 16

Easy

What is the equation of a line with slope $m = 2$ and y-intercept $b = 5$ ?

Why It Matters

Writing linear equations is a fundamental skill that connects algebra to the real world:

Business: A company charges 50 dollars per hour plus a 100 dollar service fee: $y = 50x + 100$
Science: Temperature drops 2 degrees every hour from an initial 20 degrees: $T = -2h + 20$
Finance: You save 25 dollars per week starting with 150 dollars: $S = 25w + 150$
Sports: A runner covers 8 meters per second: $d = 8t$

Mastering this skill lets you model and predict real-world relationships mathematically!

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 $C = 0.10t + 30$ .

1Try It Yourself

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 $r$ 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: $T = -5t + 80$

2Try It Yourself

A candle loses 2 centimeters of height per hour. It starts at 20 cm tall.

Write an equation for the candle's height after $h$ 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: $S = 50w + 200$

3Try It Yourself

You have 75 euros saved and add 15 euros each week from your allowance.

Write an equation for your total savings after $w$ weeks.

Step 1: Write the mathematical expression

Savings equation:

Key Takeaways

1Use $y = mx + b$ when you know the slope and y-intercept directly
2Use point-slope form $y - y_1 = m(x - x_1)$ when you know a point and slope
3Calculate slope from two points: $m = \frac{y_2 - y_1}{x_2 - x_1}$
4From tables: slope is the constant change in $y$ divided by change in $x$
5In word problems: the rate is the slope, the starting value is the y-intercept

Frequently Asked Questions

If you know the y-intercept, use slope-intercept (

y = mx + b

). If you only know a point that's not on the y-axis, point-slope (

y - y_1 = m(x - x_1)

) is easier. Both give the same line!

Glossary

Slope-intercept form: The equation $y = mx + b$ where $m$ is the slope and $b$ is the y-intercept
Point-slope form: The equation $y - y_1 = m(x - x_1)$ using slope $m$ and point $(x_1, y_1)$
Y-intercept: The point where the line crosses the y-axis; the value of $y$ when $x = 0$
Rate of change: Another name for slope; how much $y$ changes for each unit change in $x$

Formula Card

Slope-Intercept Form

y = mx + b

Use when you know the slope and y-intercept

Point-Slope Form

y - y_1 = m(x - x_1)

Use when you know a point and the slope

Slope Formula

m = \frac{y_2 - y_1}{x_2 - x_1}

Calculate slope from two points

Writing Linear Equations

Definition

Worked Examples

Common Mistakes

Interactive Visual

Linear Function Explorer

Interactive Sandbox

Interactive Grapher

Expression Calculator

History

Practice Problems

Why It Matters

Real World Applications

Cell Phone Plans

Science Experiments

Savings Goals

Key Takeaways

Frequently Asked Questions

Glossary

Formula Card

Slope-Intercept Form

Point-Slope Form

Slope Formula

More in This Topic

Introduction to Linear Functions

Slope-Intercept Form (y = mx + b)

Understanding Slope

Point-Slope Form