Linear Equations

A linear equation is one that consists of two variables and gives you a straight line when you plot the variable on a graph. It can be written in the form of:

ax + b = 0

Here x is a variable, where and b are real numbers. This is a standard form of a linear equation. If you want to solve linear equations, then please remember the facts provided below:

  • If x = y, then x + z = y + z

It means if we add a number z to both sides of the equation, it doesn’t change the equation.

  • If x = y, then x – z = y – z

In the same way, if we subtract a number z from both sides of the equation, it doesn’t change the equation.

  • If x = y, then xz= yz

In the same way, if we multiply a number z to both sides of the equation, it doesn’t change the equation either.

  • If x = y, then x / z = y / z

In the same way, if we divide a number z from both sides of the equation, it doesn’t change the equation.

Make sure to remember these facts as these form the basis in order to solve any kind of linear equations. Simply put, in solving linear equations, you need to always remember that what you do in one side of the equation should also be done on the other side of the equation as well.

Simple Steps to Solve Linear Equations

Step 1) First step in solving a linear equation is to clear the fractions by using the least common denominator by multiply it on both sides.
Step 2) Next step is to simplify the equation by combining like terms and eliminating the parenthesis.
Step 3) The first two steps will ensure the variables are pushed to one side and the constants to the other side of the equation.
Step 4) In case the coefficient of a variable is not one, then multiply or divide by a number to simplify it further to make the coefficient one.
Step 5) Next, we need to verify the answer by replacing the results we get. You will understand more clearly, if you look at the example below.

Example 1: Solve 3(x+5) = 2 (-6-x) -2x

Let’s follow the steps provided above to solve the linear equation. Since there are no fractions, we don’t need to worry about the first step. The second step is to simplify the linear equation:

3x + 15 = –12 – 2x – 2x
3x + 15 = –12 – 4x

The next step is to push the variables to one side and the constants to the other side, so we add and subtract values to both sides of the equation:

3x + 15 –15 + 4x = –12 – 4x –15 +4x
7x = -27

So, x= -27 /7

Now as per step 5, we can verify the answer by replacing the value of x in the above equation, which gives us:

3(x+5) = 2 (-6-x) -2x

When replace the value of x,

3((-27 /7) +5) = 2 (-6-(-27 /7)) -2(-27 /7)
3 (8/7) = 2 (-15/7) + (54/7)
24/ 7 = 24/7

Hence it is proved the solution is correct.

Example 2: Solve 3x + 5x + 4 – x + 7 = 88

Combine like terms:

Solve 3x + 5x + 4 – x + 7 = 88
7x + 11 = 88

Subtract 11 from both sides:

7x -11 +11 = 88 – 11
7x = 77

Divide each side by 7:

7x/7 = 77/7
x = 11

Example 3: -20y + 15 = 2 – 16y + 11

Combine like terms on each side.

-20y + 15 = 2 – 16y + 11
-20y + 15 = 13 -16y

Get the y terms on one side and all other terms on the other side:

+20y -20y +15 = 13 -16y +20y
15 = 13 + 4y

Subtract 13 from both sides:

15 – 13 = 13 – 13 +4y
2 = 4y

divide by 4 on both sides:

2/4 = 4y/4
2/4 = ½ = y