Learn on PengiBig Ideas Math, Course 3Chapter 4: Graphing and Writing Linear Equations

Lesson 7: Writing Equations in Point-Slope Form

In this Grade 8 lesson from Big Ideas Math, Course 3, Chapter 4, students learn how to write linear equations in point-slope form, y − y₁ = m(x − x₁), given a slope and a point or two points on a line. Students derive the point-slope formula by analyzing rise and run between two coordinate points, then apply it to write and convert equations into slope-intercept form.

Section 1

Point-slope form of an equation

Property

The point-slope form of an equation of a line with slope mm and containing the point (x1,y1)(x_1, y_1) is

y−y1=m(x−x1)y - y_1 = m(x - x_1)

We can use the point-slope form of an equation to find an equation of a line when we are given the slope and one point.

Examples

  • A line has a slope of 5 and passes through the point (2,8)(2, 8). Its equation in point-slope form is y−8=5(x−2)y - 8 = 5(x - 2).
  • A line has a slope of −34-\frac{3}{4} and passes through the point (−4,1)(-4, 1). Its equation is y−1=−34(x−(−4))y - 1 = -\frac{3}{4}(x - (-4)), which simplifies to y−1=−34(x+4)y - 1 = -\frac{3}{4}(x + 4).

Section 2

Writing an Equation in Point-Slope Form from a Slope and a Point

Property

To write the equation of a line in point-slope form, you use the slope mm and the coordinates of a known point (x1,y1)(x_1, y_1). The formula is:

y−y1=m(x−x1)y - y_1 = m(x - x_1)

Examples

Section 3

Writing Equations Using Two Points

Property

When given two points, first find the slope using the formula m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}. Then, choose one of the points and use the point-slope form, y−y1=m(x−x1)y - y_1 = m(x - x_1), to write the equation of the line.

Examples

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Point-slope form of an equation

Property

The point-slope form of an equation of a line with slope mm and containing the point (x1,y1)(x_1, y_1) is

y−y1=m(x−x1)y - y_1 = m(x - x_1)

We can use the point-slope form of an equation to find an equation of a line when we are given the slope and one point.

Examples

  • A line has a slope of 5 and passes through the point (2,8)(2, 8). Its equation in point-slope form is y−8=5(x−2)y - 8 = 5(x - 2).
  • A line has a slope of −34-\frac{3}{4} and passes through the point (−4,1)(-4, 1). Its equation is y−1=−34(x−(−4))y - 1 = -\frac{3}{4}(x - (-4)), which simplifies to y−1=−34(x+4)y - 1 = -\frac{3}{4}(x + 4).

Section 2

Writing an Equation in Point-Slope Form from a Slope and a Point

Property

To write the equation of a line in point-slope form, you use the slope mm and the coordinates of a known point (x1,y1)(x_1, y_1). The formula is:

y−y1=m(x−x1)y - y_1 = m(x - x_1)

Examples

Section 3

Writing Equations Using Two Points

Property

When given two points, first find the slope using the formula m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}. Then, choose one of the points and use the point-slope form, y−y1=m(x−x1)y - y_1 = m(x - x_1), to write the equation of the line.

Examples