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General algebra

Rearranging equations

  • The key point to remember when solving or re-arranging equations is to move the variable of interest to one side of the equal sign and everything else to the other.
  • Like terms are those terms which contain the same power of the same variable.
  • To move something to the other side of the equals sign you need to “undo” the operation, that is do the opposite. For example, to “undo” adding 5, you need to subtract 5 from both sides.
  • There are a number of operations which can be used to rearrange formulae. These include:
    • Adding or subtracting terms from both sides
    • Multiplying or dividing throughout
    • Changing the sign throughout
    • Raising both sides to the same power (e.g. squaring or taking the square root)
    • Taking reciprocals of both sides
    • Expanding brackets

For example: solve for \(x\): \(\displaystyle 3(x+3)=\frac{3}{8}\)

\(
\newcommand{\eqncomment}[2]{\quad \small{\text{ #2}} } 
\newcommand{\ceqns}{ \begin{array}{rcll}}
\newcommand{\ceqne}{\end{array}}
\)
 
\[
\ceqns 
3 (x+3) &=&  \frac{3}{8} \\ \\
3 (x+3) \div 3 &=&   \frac{3}{8} \div3 & \eqncomment{0.5}{divide by \(3\) on both sides,} \\ \\
x+3 &=& \frac{1}{8}\\  \\
x+3-3&=&  \frac{1}{8}- 3 & \eqncomment{0.5}{take 3 from both sides,} \\ \\
x &=& -2\frac{7}{8}
\ceqne
\]