# Writing Linear Equations

In this article, you learn how to write the equation of the lines by using their slope and one point or using two points on the line.

## Step by step guide to writing linear equations

• The equation of a line in slope intercept form is: $$\color{blue}{y=mx+b}$$
• Identify the slope.
• Find the $$y$$–intercept. This can be done by substituting the slope and the coordinates of a point $$(x, y)$$ on the line.

### Example 1:

What is the equation of the line that passes through $$(1, -2)$$ and has a slope of $$6$$?

Solution:

The general slope-intercept form of the equation of a line is $$y=mx+b$$, where m is the slope and b is the $$y$$-intercept.
By substitution of the given point and given slope, we have: $$-2=(1)(6)+b$$
So, $$b= -2-6=-8$$, and the required equation is $$y=6x-8$$.

### Example 2:

Write the equation of the line through $$(1, 1)$$ and $$(-1, 3)$$.

Solution:

Slop $$= \frac{y_{2}- y_{1}}{x_{2} – x_{1} }=\frac{3- 1}{-1- 1}=\frac{2}{-2}=-1 → m=-1$$
To find the value of b, you can use either points. The answer will be the same: $$y=-x+b$$
$$(1,1) →1=-1+b→b=2$$
$$(-1,3)→3=-(-1)+b→b=2$$
The equation of the line is: $$y=-x+2$$

### Example 3:

What is the equation of the line that passes through $$(2,–2)$$ and has a slope of $$7$$?

Solution:

The general slope-intercept form of the equation of a line is $$y=mx+b$$, where $$m$$ is the slope and $$b$$ is the $$y-$$intercept.
By substitution of the given point and given slope, we have: $$-2=(7)(2)+b$$
So, $$b= –2-14=\ -16$$, and the required equation is $$y=7x-16$$.

### Example 4:

Write the equation of the line through $$(2,1)$$ and $$(-1,4)$$.

Solution:

Slop $$= \frac{y_{2}- y_{1}}{x_{2} – x_{1} }=\frac{4- 1}{-1- 2}=\frac{3}{-3}=\ -1 → m= \ -1$$
To find the value of b, you can use either points. The answer will be the same: $$y= \ -x+b$$
$$(2,1) →1=-2+b→b=3$$
$$(-1,4)→4=-(-1)+b→b=3$$
The equation of the line is: $$y= \ -x+3$$

## Exercises

### Write the slope–intercept form of the equation of the line through the given points.

1. $$\color{blue}{through: (– 4, – 2), (– 3, 5)}$$
2. $$\color{blue}{through: (5, 4), (– 4, 3) }$$
3. $$\color{blue}{through: (0, – 2), (– 5, 3) }$$
4. $$\color{blue}{through: (– 1, 1), (– 2, 6) }$$
5. $$\color{blue}{through: (0, 3), (– 4, – 1) }$$
6. $$\color{blue}{through: (0, 2), (1, – 3) }$$

1. $$\color{blue}{y = 7x + 26}$$
2. $$\color{blue}{y = \frac{1}{9} x + \frac{31}{9}}$$
3. $$\color{blue}{y = – x – 2}$$
4. $$\color{blue}{y = – 5x – 4}$$
5. $$\color{blue}{y = x + 3}$$
6. $$\color{blue}{y = – 5x + 2}$$