## Dictionary Definition

The Merriam-Webster dictionary defines simultaneous equation as "satisfied by the same values of the variables."## A Real Example of Simultaneous Equations

It is easier to understand simultaneous equations with an example. The most common type of simultaneous equations are a set of 2 linear equations, where the unknowns are the variables**x**and

**y**:

- Both of these equations are linear. They show a straight line when plotted on a graph.
- Both of these equations have two unknowns (or variables) :
**x**and**y**. They can take different values. - Each equation has different coefficients and constants.
In the top equation, the coefficient of
**x**is**2**. The coefficient of**y**is**1**. The constant is**4**. In the bottom equation, the coefficient of**x**is**1**. The coefficient of**y**is**3**. The constant is**7**.

## How to Solve Simultaneous Equations

The solution to simultaneous equations are the values of**x**and

**y**which solve the equations simultaneously. The solution to the example above is

**x = 1**and

**x = 2**:

solving simultaneous equations

## More Real Examples of Simulaneous Equations

Simultaneous equations do not have to be 2 linear equations.## 3 (or More) Simultaneous Equations

It is possible for 3 or more linear equations to be solved simultaneously. 3 equations with 3 unknowns (**x**,

**y**and

**z**) are shown below. There are as many equations as there are unknowns (see

**Note**.) These simultaneous equations are solved by

**x = 1**,

**y = 2**and

**z = 3**.

## Linear and Quadratic Equations

It is possible for one equation to be linear and another to be a quadratic equation. Because of the quadratic equation, there will be 2 pairs of values of**x**and

**y**that solve the equations simultaneously. These equations are solved by

**x = 0**,

**y = 0**and

**x = 1**,

**y = 1**.

## What's in a Name?

Simultaneous equations are so called because the solution solves all the equations at the same time, or simultaneously.## Number of Unknowns = Number of Equations

When there are 2 sets of linear equations:- there are
**2 equations**. - there are also
**2 unknowns**(**x**and**y**).

**3**unknowns (e.g.

**x**,

**y**and

**z**), we would need

**3**equations to solve them.

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