How to Write 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.

Writing Linear Equations – 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 $$y=6x-8$$.

Writing Linear Equations – 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: $$y=-x+2$$

Writing Linear Equations – 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 $$y=7x-16$$.

Writing Linear Equations – 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 for Writing Linear Equations

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. $$\color{blue}{y = – 5x + 2}$$

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