The Equation of a Line Help

Introduction to The Equation of a Line

The equation of a line in the plane will describe—in compact form—all the points that lie on that line. We determine the equation of a given line by writing its slope in two different ways and then equating them. Some examples best illustrate the idea.

Example 1

Math Note: The form y = 3 x − 5 for the equation of a line is called the slope-intercept form . The slope is 3 and the line passes through (0, 5) (its y -intercept).

Math Note: Equation (*) may be rewritten as y − 1 = 3( x − 2). In general, the line with slope m that passes through the point ( x ₀ , y ₀ ) can be written as y − y ₀ = m ( x − x ₀ ). This is called the point-slope form of the equation of a line.

You Try It: Write the equation of the line that passes through the point (−3, 2) and has slope 4.

Example 2

You Try It: Find the equation of the line that passes through the points (2, −5) and (−6, 1).

In general, the line that passes through points ( x ₀ , y ₀ ) and ( x ₁ , y ₁ ) has equation

This is called the two-point form of the equation of a line.

Example 3

Summary:

We determine the equation of a line in the plane by finding two expressions for the slope and equating them.

If a line has slope m and passes through the point ( x ₀ , y ₀ ) then it has equation

y − y ₀ = m ( x − x ₀ ).

This is the point-slope form of a line.

If a line passes through the points ( x ₀ , y ₀ ) and ( x ₁ , y ₁ ) then it has equation

This is the two-point form of a line.

You Try It: Find the line perpendicular to 2 x + 5 y = 10 that passes through the point (1, 1). Now find the line that is parallel to the given line and passes through (1, 1).

Practice problems for this concept can be found at: Calculus Basics Practice Test.