EDEXCEL GCSE HIGHER MATHS - What are Linear, Quadratic, and Simultaneous Equations

Edexcel GCSE Higher Maths: Linear, Quadratic, and Simultaneous Equations

1. Linear Equations

A linear equation is an equation where the highest power of the variable is 1. It can be written in the form:

ax + b = 0

Where a and b are constants, and x is the variable.

Examples:

• 2x + 5 = 0
• 3x - 7 = 11
• -4x + 2 = 0

Solving Linear Equations:

To solve a linear equation, we isolate the variable on one side of the equation. We can achieve this by using inverse operations.

Example:

Solve the equation: 2x + 5 = 0

1. Subtract 5 from both sides: 2x = -5
2. Divide both sides by 2: x = -2.5

2. Quadratic Equations

A quadratic equation is an equation where the highest power of the variable is 2. It can be written in the form:

ax^2 + bx + c = 0

Where a, b, and c are constants, and x is the variable.

Examples:

• x^2 + 2x - 3 = 0
• 3x^2 - 5x + 1 = 0
• -2x^2 + 7x + 4 = 0

Solving Quadratic Equations:

There are several methods for solving quadratic equations, including:

• Factoring: Factoring the equation into two linear expressions.
• Quadratic Formula: Using the formula x = (-b ± ?(b^2 - 4ac)) / 2a
• Completing the Square: Manipulating the equation to form a perfect square trinomial.

Example:

Solve the equation: x^2 + 2x - 3 = 0

1. Factoring: (x + 3)(x - 1) = 0. Therefore, x = -3 or x = 1
2. Quadratic Formula: x = (-2 ± ?(2^2 - 4 * 1 * -3)) / (2 * 1) = (-2 ± ?16) / 2 = (-2 ± 4) / 2. Therefore, x = -3 or x = 1

3. Simultaneous Equations

Simultaneous equations are a set of two or more equations that share the same variables. The goal is to find the values of the variables that satisfy all equations simultaneously.

Types of Simultaneous Equations:

• Linear Simultaneous Equations: Both equations are linear.
• Non-Linear Simultaneous Equations: One or both equations are non-linear (e.g., quadratic, exponential).

Solving Simultaneous Equations:

• Elimination Method: Eliminate one variable by adding or subtracting the equations.
• Substitution Method: Solve one equation for one variable and substitute it into the other equation.

Example:

Solve the following simultaneous equations:

x + y = 5
2x - y = 1

Elimination Method:

1. Add the two equations: 3x = 6
2. Divide both sides by 3: x = 2
3. Substitute x = 2 into either original equation: 2 + y = 5
4. Solve for y: y = 3

Solution: x = 2 and y = 3

Key Takeaways:

• Linear equations have a highest power of 1 for the variable.
• Quadratic equations have a highest power of 2 for the variable.
• Simultaneous equations are a set of equations with shared variables.
• Mastering these equation types is essential for success in GCSE Higher Maths.

Remember to practice solving various examples of these equations to solidify your understanding and build confidence.