Year 9 Algebra Essentials: Linear Equations Mastery

1. Simplifying Algebraic Expressions

How to Simplify Algebraic Expressions

• Multiply numbers and letters together.
• Combine like terms (terms with the same letters and powers).

Example:
−2ac × 4bd = −8abcd

Example:
5ab − 8b²a + ba = 5ab − 8ab² + ab = −8ab² + 6ab

2. Expanding Brackets (Distributive Law)

How to Expand Brackets

• Multiply everything inside the bracket by what is outside.

Example:
−4(x + 7) = −4x − 28

Example:
5(x − 3) + 2(6 − x) = 5x − 15 + 12 − 2x = 3x − 3

3. Solving Equations

How to Solve Equations

• Get all variables (e.g., x's) on one side and numbers on the other.
• Perform the same operation on both sides of the equation.

Example:
3x + 2 = 17
3x = 15
x = 5

Example:
9x − 8 = 4x + 7
5x = 15
x = 3

4. Solving Inequalities

How to Solve Inequalities

Follow the same steps as solving equations, but maintain the inequality sign.

Example:
4(x + 1) ≤ 8
4x + 4 ≤ 8
4x ≤ 4
x ≤ 1

5. Substitution into Formulas

How to Substitute into Formulas

• Replace letters with their given numerical values and then calculate.

Area of a Trapezium Formula:
A = ½ (a + b) h

Example:
Given a = 3, b = 7, h = 9
A = ½ (3 + 7)(9) = ½ (10)(9) = 45

6. Rearranging Formulas (Transposition)

How to Rearrange Formulas

Isolate the desired letter or variable to make it the subject of the formula.

Example:
Given A = ½ (a + b)h

Make a the subject:
2A = (a + b)h
2A / h = a + b
a = (2A / h) − b

7. Worded Problems (Linear Equations)

How to Solve Worded Problems with Linear Equations

• Translate the problem's words into mathematical equations.
• Solve the equations step-by-step.

Example:
A rectangle is 5 times as long as it is wide. Its perimeter is 360 units.
Let width = x. Length = 5x

Perimeter = 2x + 2(5x) = 360
2x + 10x = 360
12x = 360
x = 30
Length = 5x = 150

8. Simultaneous Equations (Substitution Method)

How to Solve Simultaneous Equations by Substitution

• Substitute one equation into the other to eliminate a variable.

Example:
Given: 5x − 3y = 9 and y = 2x − 5

Substitute the second equation into the first:
5x − 3(2x − 5) = 9
5x − 6x + 15 = 9
−x = −6
x = 6

Then, substitute x = 6 back into y = 2x − 5:
y = 2(6) − 5 = 7

9. Simultaneous Equations (Elimination Method)

How to Solve Simultaneous Equations by Elimination

• Align the equations.
• Add or subtract the equations to cancel out one variable.

Example:
Given:
A + C = 160
7A + 4C = 925

Multiply the first equation by 4 to match the 'C' coefficients:
4A + 4C = 640

Now, subtract this new equation from the second original equation:
(7A + 4C) − (4A + 4C) = 925 − 640
3A = 285
A = 95

Then, substitute A = 95 back into A + C = 160:
95 + C = 160
C = 65