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Point Slope Form

Sep 15, 2022
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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) 
Point Slope Form
Point Slope Form

We can write the equation of a line using any point (x1, y1) and the slope m in point-slope form.  

parallel

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. 

Write an equation in point-slope form 

The equation in point–slope form is y+1 = 3x -9. 

parallel

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. 

graph

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

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

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

  1. Sketch the graph of the given equation.

y-1 =  5/4(x+2)

  1. 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.
table
  1. 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).
  2. Find the slope of the line that contains the points from the table
table 2
  1. 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.
Estimate the value of the y-intercept of the line.
  1. 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.
point slope
  1. 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

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

Comments:

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