**Vocabulary on simultaneous linear equations**

**1. Linear equations**

A **linear equation in two unknowns** is an equation of first degree with two unknowns.

Any linear equation can be expressed in its **standard form**

*ax + by = c*

where *a, b* and *c *are known numbers.

A **solution** of a linear equation in two unknowns is a pair of values that verify the equation (they make it true)

A linear equation in two unknowns has **infinite solutions**.

We call **system (or pair) of linear equations or simultaneous equations** in two unknowns to a collection of such equations.

*ax + by = c*

*a’x + b’y = c’*

A **solution of the system** is any pair of numbers that satisfies them simultaneously.

**Solving a pair of simultaneous equations** is to look for the values of the unknowns which make all the equations true simultaneously

### **2. Methods of solving simultaneous equations**

We now focus on various methods of solving simultaneous equations. All these methods have the same philosophy: to obtain from the two equations another equation only in one unknown. Once we solve it is easy to determine the value of the other unknown.

### **2.1. Substitution method**

** **Here it is the method of substitution:

- Solve one of the equations for one unknown in terms of the other.
- Then, substitute that in the other equation.
- That will yield one equation in one unknown, which we can solve.
- We obtain the other unknown substituting in any equation of the system

In this page you can introduce any system of linear equations, solve it by yourself and find a complete explanation about how to do it by using substitution method. Unfortunately, this algebra solver don’t use elimination method.

Here you are a video on substitution method.

**2.2. Equalizing Method**

** **This method is not explained in Anglo-Saxon countries, probably because it is a slight variation of substitution method.

** **Here it is the method of “Equalizing”:

- Solve both equations for the same unknown in terms of the other.
- Then, equalize the expressions in terms of the other.
- That will yield one equation in one unknown, which we can solve.
- We obtain the other unknown substituting in any equation of the system

Sorry, I couldn’t find a web page where the equalizing method were explained and that had interactive exercises.

** ****2.3. Elimination method**

** ****This method is also named addition method.**

** **As the name suggest, this method tries to eliminate variables until there’s only 1 variable left by adding the equations.

Here it is a general strategy for solving simultaneous equations:

- Look at the equations and try to find 2 equations which has the same coefficient (plus or minus) for the same variables.
- When the coefficients don’t have the same coefficients (plus or minus), multiply the equations so that the coefficients of that unknown are the same
- Add or subtract the equations vertically, and that unknown will cancel(we eliminate the unknown)
- We will then have one equation in one unknown, which we can solve.
- To solve for y, the other unknown substitute the value of x in one of the original equations.

This page is highly recommendable because you can find several videos on solving simultaneous equations by using elimination method, but they have got a higher level. The video below is elemental but good to improve your listening

In the page I am going to recommend you, you have interactive exercises on solving simultaneous equations by using elimination method. **It has a difficulty because they always subtract instead of add up as we do. **

**Take into account that if you subtract an equation from other you have to change every sign in the LHS and in the RHS to subtract correcly. **

Click on http://mathsfirst.massey.ac.nz/Algebra/SystemsofLinEq/EMeth.htm

Go down the page to find the exercise. Get an example and try it by yourself. If you have got any difficulty, click on the buttons below. Finally, use only example 1. *Example 2 is harder to understand for you.*