Solving Equations

Type 1: Only 1 Unknown

These are the simplest type of linear equation and can be solved easily.

Solve 6x-4 =26

Step 1: Rearrange so x's are alone on one side

\begin{aligned}(+4)\,\,\,\,\,\,\,\,\,6x-4 &=26 \\ 6x&= 30 \end{aligned}

Step 2: Divide both sides by the number before the unknown

\begin{aligned}(\div6) \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, 6x&= 30 \\ x&= 5\end{aligned}

Type 2: Unknown appears more than once

This type is similar to Type 1 but has one additional step.

Solve 12x +8 = 5x + 36

Step 1: Rearrange so that all x's are on one side

\begin{aligned}(-5x)\,\,\,\,\,\,\,\,\,12x +8 &= 5x+36 \\ 7x+8 &=36\end{aligned}

Step 2: Follow the steps for Type 1

\begin{aligned}(-8)\,\,\,\,\,\,\,\,\,7x+8 &=36 \\ (\div7)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,7x&=28 \\ x&=4\end{aligned}

Type 3: Includes brackets

This example will be very similar to Type 2, but this time containing brackets. The brackets are a barrier to our solution so we need to expand any brackets first.

Solve 3(2x-6)=2(5x+3)

Step 1:  Multiply out the brackets

\begin{aligned}3(2x-6) &= 2(5x+3)\\ 6x-18 &=10x+6\end{aligned}

Step 2: Follow the steps from Type 2

\begin{aligned}(-6)\,\,\,\,\,\,\,\,\, 6x-18 &=10x+6 \\ (-6x)\,\,\,\,\,\,\, 6x -24 &=10x \\(\div4) \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,-24 &= 4x \\ -6 &= x \end{aligned}

Type 4: Includes a fraction(s)

Fractions make things a little more complicated. You always need to try and remove the fractions first before performing any other calculations.

Solve \dfrac{4x+5}{5} = \dfrac{x+17}{3}

Step 1: Multiply out the fractions

To do this we multiply the both sides of the equation by the denominator of the fractions, first (\times5) then (\times3).

\begin{aligned}(\times5) \,\,\,\,\,\,\,\,\, \dfrac{4x+5}{5} &= \dfrac{x+17}{3} \\\\ (\times3)\,\,\,\,\,\,\,\,\,\, 4x+5 &= \dfrac{5(x+17)}{3} \\\\ \,\,\,\,\,\,\,\,\, 3(4x+5) &= 5(x+17)\end{aligned}

Step 2: Follow the steps from Type 3

First multiply out the brackets,

\begin{aligned}3(4x+5) &= 5(x+17)\\ 12x+15 &= 5x+85\end{aligned}

Then solve the equation,

\begin{aligned}(-15)\,\,\,\,\,\,\,\,\,12x+15 &= 5x+85 \\(-5x)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,12x & = 5x +70 \\(\div7)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, 7x &= 70 \\ x&=10\end{aligned}

Type 5: Squares and Square roots

Removing a square or square root is often an extra final step to solving some equations.

Example 1 – Including a square

Solve 5x^2 = 320

First we solve the equation the same as Type 1

\begin{aligned}(\div5)\,\,\,\,\,\,\,\,\,5x^2 &= 320 \\ \,\,\,\,\,\,\,\,\, x^2 &=64\end{aligned}

Finally we must perform the opposite operation to a square,  a square root.

\begin{aligned}(\sqrt{})\,\,\,\,\,\,\,\,\,x^2 &= 64 \\ \,\,\,\,\,\,\,\,\, x &= \pm8\end{aligned}

Example 2 – Including a square root

Solve 3\sqrt{x} = 15

First we solve the equation the same as Type 1

\begin{aligned}(\div3)\,\,\,\,\,\,\,\,\,3\sqrt{x} &= 15 \\ \,\,\,\,\,\,\,\,\, \sqrt{x} &=5\end{aligned}

Finally we must perform the opposite operation to a square root, a square.

\begin{aligned}(\,^2)\,\,\,\,\,\,\,\,\,\sqrt{x} &= 5 \\ \,\,\,\,\,\,\,\,\, x &=25\end{aligned}