Equations that need multiple steps in simplification before solving

The first thing you should do when solving an equation is to make certain that both sides of the equal sign are in simplified form, meaning that there is no multiplication, division, addition, or subtraction left to perform.

Let's work step-by-step through an example:        $$6x - 4 + 2x = 8 + 12$$

Here, we have like terms to combine on both sides of the equal sign. On the left side, x's must be combined with x's--on the right side, 2 numbers must be combined.

Combining 6x with 2x on the left side and 8 with 12 on the right side of the equal sign:

$$\begin{align} {\color{Red}6x} - 4 {\color{Red}+ 2x} &= {\color{Red}8 + 12} \\ 8x - 4 &= 20 \\ \end{align}$$

Now, we have an equation that is similar to ones found in previous sections.

$$\begin{align} 8x - 4 &= 20 \\ 8x - 4 + 4 &= 20 + 4 \\ 8x &= 24 \\ \frac{8x}{8} &= \frac{24}{8} \\ x &= 3 \end{align}$$

Example 1:       $$3a + 4 + 2a = -(5a -2) +4a + 6$$
Both sides of the equation need to be simplified. On the left side, like terms must be combined. On the right side, -1 must be multiplied through the parentheses and then like terms must be combined.

$$\begin{align} {\color{Red}3a} + 4 {\color{Red}+ 2a} &= {\color{Red}-1(5a -2)} +4a + 6 \\ 5a + 4 &= {\color{Red}-5a} {\color{Blue}+ 2} {\color{Red}+ 4a} {\color{Blue}+ 6} \\ 5a + 4 &= -a + 8 \end{align}$$

The variable is on both sides of the equal sign, let's add a to both sides to eliminate the variable on the right side of the equation.

$$\begin{align} 5a + 4 &= -a + 8 \\ 5a {\color{Red}+ a} + 4 &= -a {\color{Red}+ a} + 8 \\ \end{align}$$

Once we simplify by combining like terms on both sides of the equal sign, we can carry on with solving for a in the usual way:

$$\begin{align} {\color{Red}5a + a} + 4 &= {\color{Red}-a + a} + 8 \qquad \text{combine like terms on both sides} \\ 6a + 4 &= 8 \\ 6a + 4{\color{Red} - 4} &= 8 {\color{Red}- 4} \qquad \text{get the term with the variable alone by subtracting 4 from both sides} \\ 6a &= 4 \qquad \text{combine like terms on both sides of the equation} \\ \frac{6a}{\color{Red}{6}} &= \frac{4}{\color{Red}{6}} \qquad \text{divide both sides by 6} \\ a &= \frac{4}{6} = \frac{2}{3} \\ \end{align}$$

Example 2:       $$4(2r + 3) = -2(r - 5) + 2$$
In this example, there is multiplication to be done on each side of the equal sign. Using the distributive property, we multiply 4 through the terms in parentheses on the left and distribute the -2 through the terms in the parentheses on the right.

$$\begin{align} {\color{Red}4(2r + 3)} &= {\color{Blue}-2(r - 5)} + 2 \\ 8r + 12 &= -2r + {\color{Blue}10 + 2} \qquad \text{the left is in simplified form; combine like terms on the right} \\ 8r + 12 &= -2r + 12 \\ \end{align}$$

Now, both sides of the equation are completely simplified. As the variable is on both sides of the equal sign, we need to subtract 8r from both sides or add 2r to both sides of the equation. Lets add 2r.

$$\begin{align} 8r + 12 &= -2r + 12 \\ 8r {\color{Red}+ 2r} + 12 &= -2r {\color{Red}+2r} + 12 \qquad \text{add 2r to both sides of the equation} \\ 10r + 12 &= 12 \qquad \text{combine like terms on both sides of the equation} \\ 10r + 12 {\color{Red}- 12} &= 12 {\color{Red}-12} \qquad \text{subtract 12 to get the term with the r in it alone} \\ 10r &= 0 \\ \frac{10r}{\color{Red}{10}} &= \frac{0}{\color{Red}{10}} \qquad \text{ten is multiplied by r, so we divide by 10 to get r} \\ r &= 0 \\ \end{align}$$