Simultaneous equations require algebraic skills to find the values of letters within two or more equations. They are called simultaneous equations because the equations are solved at the same time.
Equations that have more than one unknown can have an infinite number of solutions. For example, \(2x + y = 10\) could be solved by:
\(x = 1\) and \(y = 8\)
\(x = 2\) and \(y = 6\)
\(x = 3\) and \(y = 4\)
However, if the solution is to work for another equation at the same time (simultaneously), then there will be just one answer. Solving simultaneous equations finds that answer.
An example of a pair of simultaneous equations is:
\(3x + y = 11\) and \(2x + y = 8\)
Solving the equations simultaneously will give the unique pair of values for \(x\) and \(y\) that work for both equations at the same time.
Solving simultaneous equations by elimination
The most common method for solving simultaneous equations is the elimination method which means one of the unknowns will be removed from each equation. The remaining unknown can then be calculated. This can be done if the coefficientThe amount that a letter has been multiplied by. In the example of 3a, the coefficient of a is 3 because 3 x a = 3a. of one of the letters is the same, regardless of sign.
Example
Solve the following simultaneous equations:
\(3x + y = 11\)
\(2x + y = 8\)
First, identify which unknown has the same coefficient. In this example this is the letter \(y\), which has a coefficient of 1 in each equation.
Either add or subtract the two equations from each other to eliminate the letter \(y\). In this example the equations will need to be subtracted from each other as \(y - y = 0\).
If the equations were added together, then \(y + y = 2y\), and so the letter \(y\) would not be eliminated.