Simultaneous equations

Equations that have more than one unknown can have an infinite number of solutions that make it true. For example, \(2x + y = 10\) could be solved by:

\(x = 1\) and \(y = 8\)
\(x = 2\) and \(y = 6\)
\(x = 3\) and \(y = 4\)

To be able to solve an equation like this, another equation needs to be used alongside it. That way it is possible to find the only pair of values that solve both equations at the same time. These are known as simultaneous equations.

Here are a couple of examples:

\(3x + y = 11\)

\(2x + y = 8\)

The values of \(x\) and \(y\) are the same in both equations. This fact can be used to help solve the two simultaneous equations at the same time and find the values of \(x\) and \(y\).

Solving simultaneous equations by elimination

The most common method for solving simultaneous equations is the elimination method. This means one letter will be removed from the equation using algebra. The remaining letter can then be calculated. This can be done if the 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 variable 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 2 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.

\(\begin{array}{ccccc} 3x & + & y & = & 11 \\ - && - && - \\ 2x & + & y & = & 8 \\ = && = && = \\ x &&& = & 3 \end{array}\)

The value of \(x\) can now be substituted into either equation to find the value of \(y\).

Substitute \(x = 3\) into either \(3x + y = 11\) or \(2x + y = 8\).

\(3x + y = 11\) when \(x = 3\)

Substitute \(x = 3\):

\(3~\mathbf{\times~3} + y = 11\)

\(9 + y = 11\)

Find the value of \(y\) using to solve equations. The inverse of adding 9 is subtracting 9, so subtract 9 from each side:

\(9 + y - 9 = 11 - 9\)

\(y = 2\)

Check the answers by substituting both values into the other original equation. If the equation balances, then the answers are correct:

\(2x + y = 8\) when \(x = 3\) and \(y = 2\).

\(2~\mathbf{\times~3} + \mathbf{2} = 8\)

\(6 + 2 = 8\)

\(8 = 8\)

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