Straight Lines and Their Equations
From plotting ordered pairs to slopes, intercepts, parallel and perpendicular lines, and intersection problems.
Learning outcomes: After completing this lesson, you should be able to plot points on a straight line, calculate its slope, work with different equation forms, identify intercepts, determine relationships between lines, and solve basic coordinate-geometry problems.
01Understanding straight lines
A straight line extends infinitely in both directions without changing direction. In coordinate geometry, its points satisfy an equation involving x and y.
Every real input x produces one output y. To see how this line is drawn, substitute several values.
| Input x | Calculation | Output y | Point |
|---|---|---|---|
| 0 | 2(0) + 1 | 1 | (0, 1) |
| 1 | 2(1) + 1 | 3 | (1, 3) |
| 2 | 2(2) + 1 | 5 | (2, 5) |
| 3 | 2(3) + 1 | 7 | (3, 7) |
Plot the four points on the coordinate plane, then connect them with a straight line and extend the line in both directions. In fact, any two distinct points determine a unique straight line.
Notice that each time x increases by 1, y increases by 2. That constant change is the slope.
Important distinction: A vertical straight line such as x = 3 is also a line, but it cannot be written in the form y = mx + b.
02Slope: steepness and direction
The slope m is the ratio of vertical change to horizontal change between two distinct points with different x-coordinates.
| Slope | Direction | Meaning |
|---|---|---|
| Positive | Rises left to right | y increases as x increases |
| Negative | Falls left to right | y decreases as x increases |
| Zero | Horizontal | y is constant |
| Undefined | Vertical | x is constant; horizontal change is zero |
Find the slope through A(−2, 1) and B(4, 7)
The slope is 1. Interchanging the two endpoints gives the same slope when both numerator and denominator are reversed.
03Slope-intercept form
For every nonvertical straight line, the equation can be expressed as:
m is the slope, and b is the y-intercept: the point (0, b).
Interpret y = −3x + 5
Slope = −3, so the line falls from left to right. Its y-intercept is (0, 5). From (0, 5), moving right by 1 and down by 3 gives another point (1, 2).
04Finding an equation from a point and a slope
When a point (x₁, y₁) and the slope m are known, use point-slope form:
Find the line of slope 2 through (3, −1)
Verification: setting x = 3 gives y = −1, so the line passes through the given point.
05Finding an equation through two points
First calculate the slope using the coordinates, then substitute one of the points into point-slope form.
Line through (1, 4) and (5, 12)
Check with both points: 2(1)+2 = 4 and 2(5)+2 = 12.
06General form and converting equations
Another common form is Ax + By + C = 0, where A and B are not both zero. When B is nonzero, solve for y to find slope and intercept.
Rewrite 2x + 3y − 6 = 0 in slope-intercept form
The slope is −2/3 and the y-intercept is (0, 2).
07x-intercepts and y-intercepts
An x-intercept occurs where the line meets the x-axis, so y = 0. A y-intercept occurs where the line meets the y-axis, so x = 0.
Find both intercepts of 2x + 3y = 12
Set y = 0: 2x = 12, so x = 6. The x-intercept is (6, 0).
Set x = 0: 3y = 12, so y = 4. The y-intercept is (0, 4).
The last expression is intercept form. It applies when both intercept values are nonzero.
08Horizontal, vertical, parallel, and perpendicular lines
| Relationship | Criterion | Example |
|---|---|---|
| Horizontal | y = c; slope 0 | y = −2 |
| Vertical | x = c; undefined slope | x = 4 |
| Parallel nonvertical lines | Equal slopes | y = 2x + 1 and y = 2x − 3 |
| Perpendicular nonvertical lines | Product of slopes = −1 | y = 2x + 1 and y = −(1/2)x + 4 |
The perpendicular-slope product rule applies only when both slopes are defined. A horizontal line is perpendicular to a vertical line. Distinct parallel lines do not intersect, and coincident lines share infinitely many points.
Line perpendicular to y = 2x − 1 through (4, 3)
Its slope is −1/2, the negative reciprocal of 2.
09Intersections of straight lines
An intersection point satisfies both line equations. Solve the two equations simultaneously.
Where do y = 2x + 1 and y = −x + 7 intersect?
The intersection point is (2, 5). It satisfies both original equations.
10Collinearity and geometric checks
Three points are collinear if they lie on one straight line. You can check this by comparing slopes, taking care with vertical lines.
Are A(1, 2), B(3, 6), and C(5, 10) collinear?
Both slopes are 2, so all three points lie on the same straight line.
11Common mistakes and how to avoid them
Mixing coordinate order
Subtract x-coordinates in the denominator and y-coordinates in the numerator in the same point order.
Confusing slope and intercept
In y = mx + b, m is the slope and b is the y-intercept.
Calling a vertical slope zero
Vertical slopes are undefined; horizontal slopes are zero.
Using negative reciprocals for parallel lines
Equal slopes mean parallel; negative reciprocal slopes mean perpendicular.
Skipping substitution checks
Verify your final equation using every given point or condition.
Forgetting coincident lines
Equal slopes do not always imply distinct lines; the intercepts may also be equal.
12Lesson review and formula reference
| Concept | Formula / test |
|---|---|
| Slope through two points | m = (y₂ − y₁)/(x₂ − x₁), x₂ ≠ x₁ |
| Slope-intercept form | y = mx + b |
| Point-slope form | y − y₁ = m(x − x₁) |
| General form | Ax + By + C = 0 |
| Horizontal / vertical | y = c / x = c |
| Parallel lines | Same slope for nonvertical lines |
| Perpendicular lines | m₁m₂ = −1 for nonvertical lines |
| Intercepts | Set y = 0 / set x = 0 |
Completion standard: You can identify the slope and intercepts of a line, construct an equation from a point or two points, determine parallel or perpendicular relationships, and solve for line intersections without confusing vertical and horizontal lines.