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# 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 (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?
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

1. Write the equation of the line in point-slope form.
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.
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
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.
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.
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).

### 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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