Lines: Slope-Intercept, Point-Slope, and Standard Form

A line is the simplest graph there is: any equation in which and appear only to the first power, with no products, roots, or fractions of the variables. Its steepness never changes, so a single number — the slope — and a single point fix the whole line.

There are three standard ways to write a line’s equation. They describe the same set of points; they just start from different information. Knowing which form fits the situation saves a lot of algebra.

The Three Forms

Slope-intercept form uses the slope and -intercept . Point-slope form uses the slope and one known point . Standard form keeps and on the same side with integer coefficients.

Slope-Intercept Form:

This is the form to graph from and to read from.

• is the slope — the coefficient of .
• is the -intercept — the constant, giving the point .

To graph : start at the -intercept , then use the slope as rise-over-run — up , right — to reach . Draw the line through the two points. No table needed.

To read a line’s slope and intercept, get it into this form first. becomes : slope , -intercept .

Point-Slope Form:

This is the form to build a line from. Whenever you know a slope and a point, substitute and you are done — no solving for .

Line with slope through :

The last step converts to slope-intercept form, which is usually the expected final answer. Point-slope is the scaffold; you rarely leave the equation in it.

Standard Form:

Conventionally , , are integers with no common factor and . Standard form is handy for two things:

• Intercepts fast. Set for the -intercept, for the -intercept — each uses only one term.
• Vertical lines. fits standard form () but cannot be written as .

Convert to standard form: multiply through by to clear the fraction, then move the -term:

Writing a Line Through Two Points

1. Slope from the two points: .
2. Point-slope with that slope and either point.
3. Simplify to the requested form.

Through and :

Using the other point, , gives , which also simplifies to . Same line, same equation.

Parallel and Perpendicular Lines

• Parallel lines have equal slopes and different -intercepts. Parallel to means slope , so for any .
• Perpendicular lines have slopes that are negative reciprocals — flip the fraction and change the sign. Perpendicular to slope is slope . The product of perpendicular slopes is always (the one exception is a horizontal and a vertical line, where one slope is undefined).

Line through perpendicular to : the new slope is , and the -intercept is , so .

Horizontal and Vertical Lines

These are the two lines that break the usual pattern:

A horizontal line has no in its equation because is free to be anything. A vertical line has no for the same reason. Trying to force into fails, which is exactly why standard form exists.

Worked Example A: Graph From Slope-Intercept Form

Graph .

Start at . Slope means down , right : next point , then . Line falls gently to the right.

Worked Example B: Equation From a Point and a Slope

Write the line with slope through in slope-intercept form.

Worked Example C: Equation From Two Points

Write the line through and .

Check with the other point: . ✓

Worked Example D: Convert Between Forms

Put into standard form.

Multiply by : . Rearrange: , then multiply by so : .

Worked Example E: Parallel and Perpendicular

Line is .

• Parallel to through : same slope , intercept → .
• Perpendicular to through : slope , then → .

Worked Example F: Horizontal and Vertical Through a Point

Give the horizontal and the vertical line through .

• Horizontal: the height is fixed at → .
• Vertical: the -value is fixed at → .

Common Mistakes to Avoid

• Reading as the slope. In , the slope is the number multiplying , not the constant.
• Not solving for before reading slope. has slope , not — rearrange to first.
• Forgetting the negative when making a perpendicular slope. Perpendicular to is ; flipping and sign-changing are both required.
• Sign slips in point-slope. becomes . Subtracting a negative point coordinate turns into addition.
• Calling a vertical line “slope zero.” Vertical is undefined slope; horizontal is zero.
• Leaving fractions in standard form. Standard form wants integer , , ; clear denominators first.
• Assuming the two points must be used in a fixed order. Either point works in point-slope, and either order works in the slope formula, as long as and subtractions match.

Where Lines Show Up Next

• Systems of equations. Two lines meet at the system’s solution — or never meet (parallel) or coincide (same line).
• Linear inequalities. Shading one side of a line is how a linear inequality is graphed.
• Functions. A non-vertical line is a linear function ; its rate of change is constant.
• Regression and modelling. A line of best fit summarises a scatter of data with one slope and one intercept.
• Calculus. The tangent line to a curve is the straight line matching the curve’s slope at a point.

Practice Problems

Work each one before opening the answer.

Problem 1. State the slope and -intercept of .

Problem 2. Rearrange into slope-intercept form and state the slope.

Problem 3. Write the line with slope through in slope-intercept form.

Problem 4. Write the line through and .

Problem 5. Write the line through and .

Problem 6. Convert to standard form.

Problem 7. Give a line parallel to that passes through .

Problem 8. Give a line perpendicular to that passes through .

Problem 9. Write the equations of the horizontal and vertical lines through .

Problem 10. Are and parallel, perpendicular, or neither?

Problem 11. A line has -intercept and -intercept . Find its equation.

Problem 12. Line passes through and is perpendicular to the line through and . Find .

Quick Reference

Every form is built on slope, the fastest points to plot are the x- and y-intercepts, and the table method for any equation is in Graphing Equations. For solving linear equations in one variable first, see Linear Equations. More Algebra lessons are available too.