Point Slope Form

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 (x_1, 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 (x_1, y₁₎ and any point (x, y) 

We can write the equation of a line using any point (x_1, 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? 
  1. Identify the point coordinates. 
  2. Identify the slope.  
  3. Input the values into the point-slope form formula: y – y₁ = m (x – x₁).  
  4. 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. 

1. Write an equation in point-slope form that describes the equation. 
2. 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

1. Write the equation in point-slope form of the line that passes through the given point with the given slope.

(3, 1); m= 2

2. Write the equation of the line in point-slope form.

3. Write an equation of the line in point–slope form that passes through the given points.

(-4, 12) and (-7, -3)

4. Sketch the graph of the given equation.

y-1 =  5/4(x+2)

5. 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.

6. 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).
7. Find the slope of the line that contains the points from the table

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

9. 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.

10. 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