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The case of solving simultaneous equations, involves equations that occur together simultaneously, and apply to specific variables, such as ** x** and

The solution of simultaneous equations, involves finding values for the variables, that satisfy all of the equations.

A system of equations is homogeneous when all equations equal

An example of two linear simultaneous equations could be:

Both of these equations on their own, can be solved for many values of

However, usually there will be only one set of

Elimination Approach

One technique of solving simultaneous equations, is to add or subtract the equations together in a
way that results in an elimination of one of the variables, with the intent that one variable being
eliminated will help give the value of the other.

With the equations seen above, they can be straight away added together as they are.

As ** y** + (-

From the remaining **3**** x** =

To get the value of

It’s often good practice to double check the value in both equations:

So the

__Examples__

__(1.1)__

Find the solutions to the following simultaneous equations.**2**** x** +

Now here adding the equations together as they sit, isn‘t going to help much, as both variables will still remain.

But, if we multiply the second equation by

**3**** x** −

Thus ** x** =

Now plugging this value into one of the original equations will give

With simultaneous linear equations such as the ones above. When we solve them, what we are actually doing is establishing the point or points where the lines of the equations intersect, that is where the lines touch each other.

In example (1.1), the * x* and

So (

This can be seen when plotting and drawing both lines.

__(1.2)__

Sometimes, simultaneous equations can have no solution.

For example,** x** +

Which means that the lines do not intersect.

Plotted on a graph the lines are actually parallel.

If a system of equations has no solution, it is said to be INCONSISTENT.

At the same time if there is a solution, then the system of equations is CONSISTENT.

Another method for solving simultaneous equations is by substitution.

Substituting one equation into the other.

__Examples__

__(2.1)__

Solve

* x* +

**Solution**

First look to re-write one of the equations in the form ** y** = …., or

The first equation

This form can then be substituted into the 2nd equation.

**2**** x** − ( -

Then to obtain the

**(2.2)**

Solve

**3*** x* +

**Solution**

First divide the first equation by **3 ** to get:* x* +

Now substitute into the 2nd equation:

* y* =

Then finally substitute the

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