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:

- Have exponents or powers

**Example:**x2 , y3 - Multiply or divide each other

**Example:**xy , ab , x/y , a/b - Be a square root

**Example:**√x , √y

Here are more examples of linear expressions:

- 2x + 4
- 4x – 2
- 5x + 3
- 4y – 8
- 4x + 10

**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.

- 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 - Since there is still a number on the “x” side, let us eliminate it by division.

9x/9 = 45/9

X = 5 - 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:**

- 4x – 2 = 2

4x -2 + 2 = 2 + 2

4x = 4

4x/4 = 4/4

X = 1 - 6x – 2 = 8

6x – 2 + 2 = 8 + 2

6x = 10

6x/6 = 10/6

X = 1 4/6

X = 1 2/3 - 7x – 5 = 14

7x – 5 + 5 = 14 + 5

7x = 19

7x/7 = 19/7

X = 2 5/7 - 6x – 1 = 17

6x – 1 + 1 = 17 + 1

6x = 18

6x/6 = 18/6

X = 3 - 8x – 14 = 2

8x – 14 + 14 = 2 + 14

8x = 16

8x/8 = 16/8

X = 2 - 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.

- Let us isolate the “x” on the other side.

2x + 4 = 10

2x + 4 – 4 = 10 -4

2x = 6 - Then let us divide both sides by 2.

2x/2 = 6/2

X = 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:**

- 2x + 4 = 5

2x + 4 – 4 = 5 - 4

2x = 1

2x/2 = ½

X = ½ - 5x + 2 = 16

5x + 2 -2 = 16 – 2

5x = 14

5x/5 = 14/5

X = 2 4/5

- 2x + 12 = 24

2x + 12 – 12 = 24 – 12

2x = 12

2x/2 = 12/2

X = 6 - 3x + 10 = 2

3X + 10 – 10 = 2 – 10

3X = -8

3X/3 = -8/3

X = 2 2/3 - 4X + 8 = 16

4X + 8 – 8 = 16 – 8

4X = 8

4X/4 = 8/4

X = 2 - 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

- 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 - 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 - 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:**

- 8x + 8y = 16

8x + 8y – 8y = 16 – 8y

8x = 16 – 8y

8x/8 = (16 – 8y)/8

X = 2 – y

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