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# Solving Multi-Step Equations

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 $x.$
Exercise

Solve $2x-1=7.$

Solution
To solve the equation, we will add $1$ to both sides, then divide by $2.$
$2x-1=7$
$2x-1+1=7+1$
$2x=8$
$\dfrac{2x}{2}=\dfrac{8}{2}$
$x=4$
Now that $x$ has been isolated, we can see that the solution is $x=4.$
info Show solution Show solution
Concept

## 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}
Concept

## 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.
Exercise

Solve $6=5x+2x-8.$

Solution
Notice there are two $x$-terms on the right-hand side of the equation. To begin, we will combine these terms.
$6=5x+2x-8$
$6=7x-8$
From here, we can solve the equation using inverse operations to isolate $x.$ Specifically, we can add $8$ to both sides, then divide by $7.$
$6=7x-8$
$6+8=7x-8+8$
$14=7x$
$\dfrac{14}{7}=\dfrac{7x}{7}$
$2=x$
$x=2$
The equation has the solution $x=2.$
info Show solution Show solution
Concept

## 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}
Exercise

Solve $x+1=\text{-} 5x-5.$

Solution
Notice in the equation that there are two variable terms — $x$ on the left-hand side and $\text{-} 5x$ on the right. We need to move one of these terms to the other side, it does not matter which. Let's move $\text{-} 5x$ to the left by adding $5x$ to the equation (the inverse operation of subtracting $5$). After that, we can combine like terms.
$x+1=\text{-} 5x-5$
$x+1+5x=\text{-} 5x-5+5x$
$6x+1=\text{-} 5$
From here, we can isolate $x$ by subtracting $1$ then dividing by $6$ on both sides.
$6x+1=\text{-} 5$
$6x+1-1=\text{-} 5-1$
$6x=\text{-} 6$
$\dfrac{6x}{6}=\dfrac{\text{-} 6}{6}$
$x=\text{-} 1$
The equation has the solution $x=\text{-} 1.$
info Show solution Show solution
Rule

## 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.

Exercise

Solve $4(\text{-} 6x+3)=12.$

Solution
To solve the equation for $x,$ we must first distribute $4$ into the parentheses.
$4(\text{-} 6x+3)=12$
$4(\text{-} 6x)+4\cdot 3=12$
$\text{-} 24x+12=12$
From here, we can subtract $12$ then divide by $\text{-} 24$ to isolate $x.$
$\text{-} 24x+12=12$
$\text{-} 24x+12-12=12-12$
$\text{-} 24x=0$
$\dfrac{\text{-} 24x}{\text{-} 24}=\dfrac{0}{\text{-} 24}$
$x=0$
The equation has the solution $x=0.$
info Show solution Show solution