EDEXCEL GCSE FOUNDATION MATHS - What are Simultaneous Equations

Edexcel GCSE Foundation Maths - What are Simultaneous Equations?

Introduction

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

Understanding Simultaneous Equations

Let's consider an example:

• Equation 1: x + y = 5
• Equation 2: 2x - y = 4

Here, the variables are x and y. We need to find values for x and y that make both equations true.

Solving Simultaneous Equations

There are two main methods for solving simultaneous equations:

1. Substitution Method:

2. Solve one equation for one variable in terms of the other.

3. Substitute this expression into the other equation.
4. Solve the resulting equation for the remaining variable.
5. Substitute the value found back into either original equation to find the value of the other variable.

Example:

• Solve Equation 1 for x: x = 5 - y
• Substitute this value of x into Equation 2: 2(5 - y) - y = 4
• Simplify and solve for y: 10 - 2y - y = 4 => 3y = 6 => y = 2
• Substitute y = 2 back into Equation 1: x + 2 = 5 => x = 3

Therefore, the solution to the system of equations is x = 3 and y = 2.

1. Elimination Method:

2. Multiply one or both equations by a constant to make the coefficients of one variable the same or opposite.

3. Add or subtract the equations to eliminate one variable.
4. Solve the resulting equation for the remaining variable.
5. Substitute the value found back into either original equation to find the value of the other variable.

Example:

• Multiply Equation 1 by 2: 2x + 2y = 10
• Add this equation to Equation 2: (2x + 2y) + (2x - y) = 10 + 4 => 4x + y = 14
• Solve for x: 4x + 2 = 14 => 4x = 12 => x = 3
• Substitute x = 3 back into Equation 1: 3 + y = 5 => y = 2

Therefore, the solution to the system of equations is x = 3 and y = 2.

Real-World Applications

Simultaneous equations are used to model and solve various real-world problems, including:

• Mixture problems: Determining the amount of different ingredients needed to create a desired mixture.
• Cost and revenue problems: Analyzing the relationship between production costs, selling prices, and profit.
• Speed, distance, and time problems: Finding the speed or time of travel based on given information.

Summary

Simultaneous equations are a powerful tool for solving systems of equations that involve multiple variables. The substitution and elimination methods are two commonly used techniques to find the solution. By understanding these methods, you can effectively solve problems involving simultaneous equations in various real-world scenarios.