LINEAR EQUATIONS

1. Definition

A linear equation is an equation in which the highest power of the variable is 1.

Example: 2x + 3 = 9

2. Linear Equations in One Variable

General form: ax + b = c

3. Solving Linear Equations

note:

A value changes sign when it crosses an equal sign.

Example: 2x - 10 = 0

2x = 10

A Negative sign (-) becomes Positive when it crosses an equal sign
A Positive sign (+) becomes Negative when it crosses an equal sign

4. Examples

Find the value of x in each equation below:

(i) x + 5 = 12

x = 12 - 5

x = 7

(ii) 3x = 15

x = 15 3

x = 5

(iii) 2x - 4 = 10

2x = 10 + 4

2x = 14

x = 7

(iv)Equations with Variables on Both Sides

When there are variables on both sides,collect like terms.

3x + 2 = x + 10

3x - x = 10 - 2

2x = 8

x = 4

(v)Equations With Fractions

When there is a fraction,multiply each term of the equation by the LCM to eliminate the denominator.

x 2 + 3 = 7

The LCM of the denominators : "2", "1" and "1" is "2"

Therefore, each term is multiplied by 2.

(x)2 2 + (3) 2= (7) 2

X + 6 = 14

X = 14 - 6

x = 8

(vi)Equations With Brackets

Simply open up the bracket by multiplying

2(x + 3) = 10

2x + 6 = 10

2x = 10 - 6

2x = 4

x = 2

5. Word Problems

(i)A number plus 6 is 15.Find the number.

Step 1: Let the number be x.

x + 6 = 15

x = 15 - 6

x = 9

(ii)Three times a number minus 5 is 16.What's the number ?

Step 1: Let the number be x.

3x - 5 = 16

3x = 16 + 5

3x = 21

x = 7

(iii) A number increased by 5 is 12. Find the number.

Step 1: Let the number be x.

This defines the unknown.

Step 2: Form the equation.

x + 5 = 12

This represents the statement.

Step 3: Solve for x.

x = 12 - 5

This isolates the variable.

Step 4: Simplify.

x = 7

This gives the answer.

(iv) A number decreased by 3 is 10. Find the number.

Step 1: Let the number be x.

This defines the unknown.

Step 2: Form the equation.

x - 3 = 10

This represents the statement.

Step 3: Solve for x.

x = 10 + 3

This isolates the variable.

Step 4: Simplify.

x = 13

This gives the answer.

(v) Three times a number is 21. Find the number.

Step 1: Let the number be x.

This defines the unknown.

Step 2: Form the equation.

3x = 21

This represents the statement.

Step 3: Solve for x.

x = 21 ÷ 3

This isolates the variable.

Step 4: Simplify.

x = 21 3

x = 7

This gives the answer.

(vi) A number divided by 4 is 6. Find the number.

Step 1: Let the number be x.

This defines the unknown.

Step 2: Form the equation.

x ÷ 4 = 6

x 4 = 6

This represents the statement.

Step 3: Solve for x.

x = 6 × 4

This isolates the variable.

Step 4: Simplify.

x = 24

This gives the answer.

(vii) The sum of a number and 9 is 20. Find the number.

Step 1: Let the number be x.

This defines the unknown.

Step 2: Form the equation.

x + 9 = 20

This represents the statement.

Step 3: Solve for x.

x = 20 - 9

This isolates the variable.

Step 4: Simplify.

x = 11

This gives the answer.

(viii) The difference between a number and 4 is 8. Find the number.

Step 1: Let the number be x.

This defines the unknown.

Step 2: Form the equation.

x - 4 = 8

This represents the statement.

Step 3: Solve for x.

x = 8 + 4

This isolates the variable.

Step 4: Simplify.

x = 12

This gives the answer.

(ix) Twice a number increased by 3 is 15. Find the number.

Step 1: Let the number be x.

This defines the unknown.

Step 2: Form the equation.

2x + 3 = 15

This represents the statement.

Step 3: Solve for x.

2x = 15 - 3

2x = 12

This simplifies the equation.

Step 4: Isolate the variable.

x = 12 ÷ 2

x = 12 2

Step 5: Simplify.

x = 6

This gives the answer.

(x) Five more than twice a number is 17. Find the number.

Step 1: Let the number be x.

This defines the unknown.

Step 2: Form the equation.

2x + 5 = 17

This represents the statement.

Step 3: Solve for x.

2x = 17 - 5

2x = 12

This simplifies the equation.

Step 4: Isolate the variable.

x = 12 ÷ 2

x = 12 2

Step 5: Simplify.

x = 6

This gives the answer.

(xi) The sum of three consecutive numbers is 30. Find the numbers.

Step 1: Let the first number be x.

This defines the starting value.

Step 2: Represent the other numbers.

x, x + 1, x + 2

This shows consecutive numbers.

Step 3: Form the equation.

x + (x + 1) + (x + 2) = 30

This represents the statement.

Step 4: Solve for x.

3x + 3 = 30

3x = 30 - 3

3x = 27

Step 5: Simplify the equation to find the value of x.

3x = 27

x = 27 ÷ 3

x = 27 3

x = 9

This is the first number.

Step 6:Find the other two numbers.

Since x = 9 , then ;

x + 1 = 10

x + 2 = 11

Final Answer :

9, 10, 11

(xii) A number plus its double is 18. Find the number.

Step 1: Let the number be x.

Since the number is x , then its double is 2x.

Step 2: Form the equation.

x + 2x = 18

This represents the statement.

Step 3: Simplify.

3x = 18

This combines like terms.

Step 4: Solve for x.

x = 18 ÷ 3

x = 18 3

x = 6

Final answer:

x = 6

Sample Questions

Attempt the following Questions:

(1.) x + 5 = 12

(2.) 10 + x = 25

(3.) x - 8 = 10

(4.) 3x = 12

(5.) x 2 = 5

Check The Answers Below:

Take Note:

Equality Matters: A linear equation contains an equals sign (=), showing that the expressions on both sides have the same value.
Keep the Equation Balanced: Whatever operation you perform on one side of an equation must also be performed on the other side.
Isolate the Variable: The main goal is to get the unknown variable by itself on one side of the equation.
Use Opposite Operations: Addition is undone by subtraction, subtraction by addition, multiplication by division, and division by multiplication.
Check Your Answer: Substitute the value you find back into the original equation. Both sides should have the same value.

LINEAR EQUATIONS

1. Definition

2. Linear Equations in One Variable

3. Solving Linear Equations

4. Examples

Find the value of x in each equation below:

5. Word Problems

Sample Questions

Check The Answers Below:

Answers:

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