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Line

Line

The geometric figure formed by two points. A line is the straight path connecting two points and extending beyond the points in both directions.

 

 

See also

Equation of a line

Key Formula

y=mx+by = mx + b
Where:
  • yy = the y-coordinate of any point on the line
  • mm = the slope (steepness) of the line
  • xx = the x-coordinate of any point on the line
  • bb = the y-intercept, where the line crosses the y-axis

Worked Example

Problem: Two points A(1, 2) and B(3, 6) define a line. Find the equation of this line in slope-intercept form.
Step 1: Calculate the slope using the two points.
m=y2−y1x2−x1=6−23−1=42=2m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{6 - 2}{3 - 1} = \frac{4}{2} = 2
Step 2: Substitute the slope and one point into y = mx + b to find b. Using point A(1, 2):
2=2(1)+b  ⟹  b=02 = 2(1) + b \implies b = 0
Step 3: Write the equation of the line.
y=2xy = 2x
Answer: The line through A(1, 2) and B(3, 6) has the equation y = 2x. This line passes through the origin and extends infinitely in both directions.

Frequently Asked Questions

What is the difference between a line, a ray, and a line segment?
A line extends infinitely in both directions and has no endpoints. A ray has one endpoint and extends infinitely in one direction. A line segment has two endpoints and a finite length. All three are straight and one-dimensional.
How many points do you need to define a line?
Exactly two distinct points define a unique line. Through any single point, infinitely many lines can pass. But once you pick two different points, only one straight line passes through both of them.

Line vs. Line Segment

A line extends infinitely in both directions with no endpoints. A line segment has two definite endpoints and a measurable, finite length.

Why It Matters

Lines are one of the most fundamental objects in all of geometry and algebra. The equation of a line is central to graphing, solving systems of equations, and modeling real-world relationships like speed, cost, and growth. Nearly every branch of mathematics—from coordinate geometry to calculus—builds on the concept of a line.

Common Mistakes

Mistake: Confusing a line with a line segment, thinking a line has two endpoints and stops.
Correction: A line has no endpoints and extends forever in both directions. When you draw one on paper, the arrows at each end indicate it continues infinitely. A line segment is the part between two endpoints.
Mistake: Believing that two points merely 'suggest' a line, rather than uniquely determining one.
Correction: Two distinct points always define exactly one line. This is a foundational postulate in Euclidean geometry: through any two distinct points, there exists one and only one line.

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