Solving equations with unknowns on both sides

Some equations have on each side of the equals sign, for example \(4 (k + 7) = 12k - 4\).

Solve this equation by rearranging all the variables onto one side of the equation and all numbers onto the other side. The easiest way to do this is usually by moving the unknown with the smallest coefficient in the equation (the variable with the smallest number in front of it).

Example

Solve \(4(k + 7) = 12k - 4\).

Expand the bracket:

\(4k + 28 = 12k - 4\)

Decide which of the unknowns has the smaller number in front of it. 4 is less than 12 so subtract \(4k\) from both sides.

\(\begin{array}{ccc} 4k + 28 & = & 12k - 4 \\ -4k && -4k \end{array}\)

\(28 = 8k - 4\)

Isolate the term \(8k\) on the right hand side by adding 4 to each side:

\(\begin{array}{ccc} 28 & = & 8k - 4 \\ +4 && +4 \end{array}\)

\(32 = 8k\)

Isolate \(k\) by dividing both sides by 8:

\(\begin{array}{ccc} 32 & = & 8k \\ \div 8 && \div 8 \end{array}\)

\(4 = k\)

Substituting \(k = 4\) back into the original equation gives:

Left hand side: \(4k + 28 = 4 \times 4 + 28 = 16 + 28 = 44\)

Right hand side: \(12k - 4 = 12 \times 4 - 4 = 48 -4 = 44\)

The equation balances, so \(k = 4\) is the correct solution.

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Solving linear equations - AQASolving equations with unknowns on both sides