In this chapter, you will learn all linear equations.

Linear equation is a form of mathematical expression that results in a straight line. It has an equal sign and linear expressions. Linear expression is a mathematical presentation that performs the action of addition, subtraction, multiplication and division.

There are various ways to write and express linear equation. It consists of a constant (example “5” or “c”) and a variable. A variable is any letter that is usually represented by “x” and “y” though any letter would be appropriate too.

Variables in the linear expressions never:

1. Have exponents or powers
   Example: x2 , y3
2. Multiply or divide each other
   Example: xy , ab , x/y , a/b
3. Be a square root
   Example:√x , √y

Here are more examples of linear expressions:

1. 2x + 4
2. 4x – 2
3. 5x + 3
4. 4y – 8
5. 4x + 10

How to solve linear equations:

Example:

9x – 43 = 2

Since it is an “equation”, the result of the expression on the left would be “equal” to the equation on the right.

1. Let us isolate the “x” in order for us to know its value.
   Based on the equation above let us first add 43 on both sides of the equation.
   9x – 43 + 43 = 2 + 43
   9x = 45
2. Since there is still a number on the “x” side, let us eliminate it by division.
   9x/9 = 45/9
   X = 5
3. Now that we know the value of “x”, we can check our answer by substituting the value of “x” to the equation.
   9*5 – 43 = 2
   45 – 43 = 2
   2 = 2

As we can see, the result of the expression on the left is the same as on the right. Thus, our answer is correct!

Exercises:

1. 4x – 2 = 2
   4x -2 + 2 = 2 + 2
   4x = 4
   4x/4 = 4/4
   X = 1
2. 6x – 2 = 8
   6x – 2 + 2 = 8 + 2
   6x = 10
   6x/6 = 10/6
   X = 1 4/6
   X = 1 2/3
3. 7x – 5 = 14
   7x – 5 + 5 = 14 + 5
   7x = 19
   7x/7 = 19/7
   X = 2 5/7
4. 6x – 1 = 17
   6x – 1 + 1 = 17 + 1
   6x = 18
   6x/6 = 18/6
   X = 3
5. 8x – 14 = 2
   8x – 14 + 14 = 2 + 14
   8x = 16
   8x/8 = 16/8
   X = 2
6. 15x – 7 = 3
   15x – 7 + 7 = 3 + 7
   15x = 10
   15x/15 = 10/15
   X = 2/3

Since we are done with the subtraction, now let us try addition.

Example:

2x + 4 = 10

Basically, the steps are the same with the first example.

1. Let us isolate the “x” on the other side.
   2x + 4 = 10
   2x + 4 – 4 = 10 -4
   2x = 6
2. Then let us divide both sides by 2.
   2x/2 = 6/2
   X = 3
3. Finally, we can check if our answer is correct by substituting the value of our solved “x”
   2*3 + 4 = 10
   6 + 4 = 10
   10 = 10

Exercises:

1. 2x + 4 = 5
   2x + 4 – 4 = 5 - 4
   2x = 1
   2x/2 = ½
   X = ½
2. 5x + 2 = 16
   5x + 2 -2 = 16 – 2
   5x = 14
   5x/5 = 14/5
   X = 2 4/5
   
3. 2x + 12 = 24
   2x + 12 – 12 = 24 – 12
   2x = 12
   2x/2 = 12/2
   X = 6
4. 3x + 10 = 2
   3X + 10 – 10 = 2 – 10
   3X = -8
   3X/3 = -8/3
   X = 2 2/3
5. 4X + 8 = 16
   4X + 8 – 8 = 16 – 8
   4X = 8
   4X/4 = 8/4
   X = 2
6. 5X + 3 = 8
   5X + 3 – 3 = 8 – 3
   5X = 5
   5X/5 = 5/5
   X = 1

How about if there are more than one variable in the equation? How would we solve it?

Example:

2x – 2y = 4

1. Let us isolate the “x” first by moving the other variable to the other side.
   2x – 2y = 4
   2x – 2y + 2y = 4 + 2y
   2x = 4 + 2y
2. Then we divide each side by 2 to come up with only “x” on the other side.
   2x = 4 + 2y
   2x/2 = (4 + 2y)/2
   X = 2 + y
3. Now, we substitute the value of our solved “x” to check if our answer is correct.
   2x – 2y = 4
   2*(2 + y) – 2y = 4
   When you look at the equation above, it already has a single variable “y”
   2*(2 + y) – 2y = 4
   4 + 2y – 2y = 4
   4 = 4

Try this:

1. 8x + 8y = 16
   8x + 8y – 8y = 16 – 8y
   8x = 16 – 8y
   8x/8 = (16 – 8y)/8
   X = 2 – y