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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 – y_{1} = m(x – x_{1})

Where *m* is the slope and (*x*_{1, }y_{1}) 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, }y_{1) }and any point (x, y)

We can write the equation of a line using any point (x_{1, }y_{1}) and the slope *m* in point-slope form.

y – y_{1} = m (x – x_{1})

- 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 – y
_{1}= m (x – x_{1}). - 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.

=y_{2}−y_{1} / x_{2}−x_{1}

= 9−4 / 6−(−4)

= 5 / 10

=1 / 2

Use slope and one point to write the equation.

y – y_{1} = m(x – x_{1})

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 (x_{1}, y_{1})_{ }is (2, 10)

Slope m =3

Equation is y – y_{1} = m(x – x_{1})

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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