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For example, the simultaneous equations \(3a + 2b = 17\) and \(4a - b = 30\) have no common coefficient as the coefficients of \(a\) are 3 and 4, and the coefficients of \(b\) are 2 and -1.

\(\begin{array}{rrrrr} 3a & + & 2b & = & 17 \\ 4a & - & b & = & 30 \end{array}\)

Create a common coefficient for either \(a\) or \(b\). In this case, making a common coefficient for \(b\) will be easier, as \(-b\) can be doubled to create \(-2b\), which will be a common coefficient throughout the equations.

Multiply the bottom equation to create a common coefficient of \(2b\).

\(\begin{array}{rrrrr} 3a & \mathbf{+} & \mathbf{2}b & = & 17 \\ 8a & \mathbf{-} & \mathbf{2}b & = & 60 \end{array}\)

These equations can now be used to find the values of \(a\) and \(b\).

\(\begin{array}{ccccc} 3a & + & 2b & = & 17 \\ + && + && + \\ 8a & - & 2b & = & 60 \\ = && = && = \\ 11a &&& = & 77 \\ \div 11 &&&& \div 11 \\ a &&& = & 7 \end{array}\)

Substitute the value of \(a\) into one of the original equations to find the value of \(b\).

Using \(3a + 2b = 17\) with \(a = 7\), gives \(21 + 2b = 17\), so \(b = -2\).

Therefore, the solution is \(a = 7\) and \(b = -2\).

It is a good idea to check this by using the other equation, \(4a – b = 30\): \(4a – b\) with \(a = 7\) and \(b = -2\) gives \(28 – (-2) = 30\), which is correct.