Equations of Lines in Coordinate Geometry

Learn how to write and interpret equations of lines using slope-intercept, point-slope, and standard forms.

Intermediate25 minLesson
Printable worksheet

Definition

The equation of a line describes all the points (x,y)(x, y) that lie on that line. There are three main forms:
1. Slope-Intercept Form:
y=mx+by = mx + b
where mm is the slope and bb is the y-intercept.
2. Point-Slope Form:
y−y1=m(x−x1)y - y_1 = m(x - x_1)
where mm is the slope and (x1,y1)(x_1, y_1) is a point on the line.
3. Standard Form:
Ax+By=CAx + By = C
where AA, BB, and CC are integers, and AA is usually positive.

Try it now

What is the slope in the equation y=5x−3y = 5x - 3?

Worked Examples

Write the equation of a line with slope 33 and y-intercept −2-2.
1

Identify the form to use

We have slope and y-intercept, so use slope-intercept form: y=mx+by = mx + b → Use y=mx+by = mx + b
2

Substitute the slope

m=3m = 3, so y=3x+by = 3x + b
3

Substitute the y-intercept

b=−2b = -2, so y=3x+(−2)y = 3x + (-2) → y=3x−2y = 3x - 2

Common Mistakes

Confusing slope and y-intercept in y=mx+by = mx + b

Why it's wrong: Students sometimes think the first number is the y-intercept. In y=3x+5y = 3x + 5, the slope is 33 (coefficient of xx) and the y-intercept is 55 (constant term).

Correct: Remember: mm comes before bb in the formula, and mm is multiplied by xx. The y-intercept is the standalone number.

Incorrect slope calculation: x2−x1y2−y1\frac{x_2 - x_1}{y_2 - y_1}

Why it's wrong: Students flip the formula, putting xx on top instead of yy.

Correct: Slope = rise over run = y2−y1x2−x1\frac{y_2 - y_1}{x_2 - x_1}. The yy values (vertical change) go on top.

Sign errors in point-slope form

Why it's wrong: When the point has a negative coordinate like (−3,2)(-3, 2), students write y−2=m(x−−3)y - 2 = m(x - -3) incorrectly as y−2=m(x−3)y - 2 = m(x - 3).

Correct: With point (−3,2)(-3, 2): x−(−3)=x+3x - (-3) = x + 3, so the equation is y−2=m(x+3)y - 2 = m(x + 3).

Forgetting to distribute in point-slope form

Why it's wrong: When expanding y−5=2(x−3)y - 5 = 2(x - 3), students write y−5=2x−3y - 5 = 2x - 3 instead of y−5=2x−6y - 5 = 2x - 6.

Correct: Distribute the slope to BOTH terms inside the parentheses: 2(x−3)=2x−62(x - 3) = 2x - 6.

Watch a video explanation

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How to Find the Equation of a Line | Step-by-Step Examples & Easy Guide

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Interactive Visual

Function graph

-5-4-3-2-11234567-5-4-3-2-1123456780xy(-1, 4)(3, -2)(2, 5)y = -1.5x + 2.5y = -4x + 13
  • Line through (-1, 4) and (3, -2): y = -1.5x + 2.5
  • Line through (2, 5) and (3, 1): y = -4x + 13

Linear Function Explorer

y = 3x - 2
Slope (m)3
Y-Intercept (b)-2
b
run
rise
xy
-2-8
-1-5
0-2
11
24

Interactive Sandbox

Interactive Grapher

-10-50510-10-50510xy

Tap or hover over the graph to read off values.

Try these examples:

y = 3x - 2

m=3, b=-2

Expression Calculator

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Practice Problems

16 problems
Problem 1 of 16
Easy
What is the slope in the equation y=5x−3y = 5x - 3?

Why It Matters

Equations of lines are fundamental in mathematics and have countless real-world applications:
  • Business: Predicting costs, revenue, and profit based on production levels
  • Science: Describing relationships between variables (temperature vs. altitude, speed vs. time)
  • Engineering: Designing ramps, roads, and structures with specific inclines
  • Economics: Modeling supply and demand, inflation trends, and growth rates
Understanding line equations allows you to make predictions, analyze trends, and solve problems involving constant rates of change.

Real World Applications

Cell Phone Plans

Phone plans often have a fixed monthly fee plus a cost per minute or gigabyte used.

Example:

A plan costs 20 dollars per month plus 0.05 dollars per text message. The equation is C=0.05t+20C = 0.05t + 20, where CC is the total cost and tt is the number of texts.

1Try It Yourself

A streaming service charges 8 euros per month plus 2 euros per movie rented.

Write the equation and find the cost for renting 5 movies.

Step 1: Write the mathematical expression

Write the cost equation CC in terms of movies mm:

Temperature Conversion

The relationship between Celsius and Fahrenheit is linear.

Example:

The formula F=95C+32F = \frac{9}{5}C + 32 converts Celsius to Fahrenheit. The slope 95\frac{9}{5} means each degree Celsius equals 1.81.8 degrees Fahrenheit.

2Try It Yourself

Water boils at 100°C100°C.

What is the boiling point in Fahrenheit?

Step 1: Write the mathematical expression

Use F=95C+32F = \frac{9}{5}C + 32 with C=100C = 100:

Taxi Fare

Taxi companies typically charge a base fare plus a rate per kilometer.

Example:

If a taxi charges 3 euros to start and 1.50 euros per kilometer, the fare equation is F=1.5d+3F = 1.5d + 3, where dd is distance in kilometers.

3Try It Yourself

A ride costs 18 euros total.

How far was the trip?

Step 1: Write the mathematical expression

Solve 18=1.5d+318 = 1.5d + 3 for dd:

Key Takeaways

  • 1Slope-intercept form y=mx+by = mx + b shows the slope mm and y-intercept bb directly
  • 2Point-slope form y−y1=m(x−x1)y - y_1 = m(x - x_1) is useful when you know a point and the slope
  • 3Standard form Ax+By=CAx + By = C uses integer coefficients with AA usually positive
  • 4To find an equation from two points: first calculate slope, then use point-slope form
  • 5All three forms describe the same line - choose the most convenient for your situation

Frequently Asked Questions

Use slope-intercept form when you need to graph quickly or identify slope/y-intercept. Use point-slope form when given a point and slope. Use standard form when working with systems of equations or when integer coefficients are required.
Use slope-intercept form when you need to graph quickly or identify slope/y-intercept. Use point-slope form when given a point and slope. Use standard form when working with systems of equations or when integer coefficients are required.
Convert both equations to the same form (usually slope-intercept). If they have the same slope and y-intercept, they represent the same line.
If slope is zero, the line is horizontal: y=by = b. If slope is undefined (vertical line), the equation is x=ax = a, which cannot be written in slope-intercept form.

Glossary

Slope-intercept form
The equation y=mx+by = mx + b where mm is slope and bb is y-intercept
Point-slope form
The equation y−y1=m(x−x1)y - y_1 = m(x - x_1) using a point (x1,y1)(x_1, y_1) and slope mm
Standard form
The equation Ax+By=CAx + By = C where AA, BB, CC are integers
Y-intercept
The point where the line crosses the y-axis, written as (0,b)(0, b)
Slope
The ratio riserun=y2−y1x2−x1\frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1} measuring steepness
Today's challengeOne problem, three tries

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