Simultaneous equations represent two or more functions in a Cartesian plane and solving them is their point of intersection.
There are three types of simultaneous equations we can have where there is one solution, no solution or infinite solutions.
There are two main methods used to solve simultaneous equations one is by elimination and the other is through matrices.
Elimination is done by making the coefficients of a variable the same across two equations and then minusing the two equations to eliminate the variable for example:
Equation 1 ×2
Equation 3-Equation 2
Sub in x=1 into equation 1
This gets more complicated with more variables but a few more steps of elimination will allow you to break the equation down to one variable. As long as you have as many equations as variables you can find the solution.
The other method is the matrix method. Matrices can be used to solve simultaneous equations and are the easiest methods if you are able to use a calculator.
The basic form of simultaneous equations is:
To solve the matrix we need to A inverse. Once we find A inverse we multiply both sides:
Since IX multiplies out to just X
Determining the inverse matrix gets very complicated for any matrices larger than a 2 by 2 matrix, however it can be easily calculated by using your calculator.
Lets look at the same equations as before:
But now in the matrix form:
How do we find the inverse? First we need the determinant.
This is the same as our result earlier. This time the method took a lot longer however if we had a calculator you can use that to calculate the inverse of A and to do the matrix multiplication. It is also essential to use a calculator to solve any set of simultaneous equations with more than three equations if you wish to use the matrix method.
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