# Solving Multi-Step Equations

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Most equations will require two or more steps to solve. The steps used when solving two-step equations are still inverse operations and the Properties of Equality. When solving, begin as far away from the variable as possible. Consider the following equation. $2x-1=7$

Notice that the variable is undergoing two operations. Namely, it is multiplied by 2, then decreased by 1. Both operations must be undone to isolate $x.$## Multi-Step Equations

In addition to requiring more than one step, some equations contain distinct elements. Below are examples of each.

$\begin{aligned} \textbf{Combining Like Terms:} & \quad 6=5x+2x-8 \\ \textbf{Variable Terms on Both Sides:} & \quad x+1=\text{-} 5x-5\\ \textbf{Distributive Property:} & \quad 4(\text{-} 6x+3)=12 \end{aligned}$## Combining Like Terms

When simplifying algebraic expressions, it is only possible to combine (add or subtract) like terms. $\begin{gathered} 7y+2x-3y-2+x+5 \end{gathered}$ This example expression contains three sets of like terms: $x\text{-terms},$ $y\text{-terms},$ and $\text{constants}.$

To simplify the expression, the terms should first be rearranged such that like terms are grouped together. Then, the like terms can be combined by adding or subtracting the constants as well as adding or subtracting the coefficients of the variables. $\begin{aligned} \colorbox{#99ffbb}{$+7y-3y$} &\quad + & \colorbox{#b3e6ff}{$+2x+x$} &\quad + & \colorbox{#ffb3b3}{$\text{-}2+5$} \\ {\color{#009600}{4y}} \quad\ &\quad + & {\color{#0000FF}{3x}} \quad\ &\quad + & {\color{#FF0000}{3}}\quad \end{aligned}$

This is the simplest form of the expression.## Variables on Both Sides

When an equation has variable terms on both sides, it is necessary to transfer them to one side using inverse operations. Once all variable terms are on the same side, they can be combined.

$\begin{aligned} 3x&=x-2\\ 3x{\color{#0000FF}{-x}}&=x{\color{#0000FF}{-x}}-2\\ 2x&=\text{-} 2 \end{aligned}$## Distributive Property

The Distributive Property can be used to simplify expressions with parentheses. The factor outside the parentheses is multiplied, or distributed,

to every term inside.

## Exercises

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