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Key Concepts
- Understand point slope form of a linear equation
- Write an equation in point slope form
- Sketch the graph of a linear equation in point slope form
- Apply linear equations
Point–Slope form
The point-slope form of a linear equation is
y – y1 = m(x – x1)
Where m is the slope and (x1, y1) is the specific point and (x, y) is any point on the line.
Understand point-slope form of a linear equation
1. How can you write the equation of a line using any points on a line?
- Use the slope formula to find the slope using the specific point (x1, y1) and any point (x, y)


We can write the equation of a line using any point (x1, y1) and the slope m in point-slope form.
y – y1 = m (x – x1)
- How to find the equation of a line with slope and coordinates of a point?
- Identify the point coordinates.
- Identify the slope.
- Input the values into the point-slope form formula: y – y1 = m (x – x1).
- Simplify to get the general equation.
Write an equation in point-slope form
1. Write the equation for the line that passes through point (3, 1) with a slope of 3.
Solution:
The slope and a point on the line are known, so use point-slope form.

The equation in point–slope form is y+1 = 3x -9.
2. Write an equation for a line that passes through the following points (-4, 4) and (6, 9)
Solution:
Find the slope of the line using the two given points.
=y2−y1 / x2−x1
= 9−4 / 6−(−4)
= 5 / 10
=1 / 2
Use slope and one point to write the equation.
y – y1 = m(x – x1)
y-4 = 1 / 2 (x+4)
y-4 = 1 / 2x +2
The equation in point –slope form is y-4 = 1 / 2x +2
Sketch the graph of a linear equation in point-slope form
Example 1:
What is the graph of y−2= 𝟐 / 𝟑 (x+2)?
Solution:
Step 1:
Identify a point on the line from the equation and plot it.
y−2= 2 / 3 (x+2)
The point is (-2, 2)
Step 2:
Use the slope to plot a second point.
m = 2 / 3
Move 2 units up and 3 units to the right and draw another point (1, 4).
Step 3: Sketch a line through the points.

Apply linear equations
Example:
Paul wants to place an ad in the newspaper. The newspaper charges $10 for the first 2 lines of text and $3 for each additional line of text.
- Write an equation in point-slope form that describes the equation.
- Find the cost of an ad that has 8 lines.
Solution:
1. Write an equation in point slope form that describes the equation.
Points on the line (x1, y1) is (2, 10)
Slope m =3
Equation is y – y1 = m(x – x1)
y-10 = 3 (x-2)
2. Find the cost of an ad that has 8 lines.
y-10 =3 (x-2)
y-10 = 3(8-2)
y – 10 = 24-6
y –10 = 18
y = 28
The cost of an ad that has 8 lines is $28.
Exercise
- Write the equation in point-slope form of the line that passes through the given point with the given slope.
(3, 1); m= 2
- Write the equation of the line in point-slope form.

- Write an equation of the line in point–slope form that passes through the given points.
(-4, 12) and (-7, -3)
- Sketch the graph of the given equation.
y-1 = 5/4(x+2)
- Write an equation of the line in point–slope form that passes through the given points in each table. Then write the equation in slope-intercept form.

- Write the slope-intercept form of the equation of the line through the given points using point-slope form through: (3, −3) and (0, −5).
- Find the slope of the line that contains the points from the table

- Use the graph of the line shown.
- Write a point-slope form of the equation for the line shown.
- Estimate the value of the y-intercept of the line.

- A railway system on a hillside moves passengers at a constant rate to an elevation of 50 m. The elevation of a train is given for 2 different locations. Write an equation in point-slope form to represent the elevation of the train in terms of the train.

- Write the slope-intercept form of the equation of the line through the given points using point-slope form through: (3, 1) and (-5, −2).
Concept Map

What have we learned
- Understand point slope form of a linear equation
- Write an equation in point slope form
- Find the slope of the line using the two given points.
- Sketch the graph of a linear equation in point slope form
- Apply linear equations
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