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. 2x1=7 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.x.
Exercise

Solve 2x1=7.2x-1=7.

Solution
To solve the equation, we will add 11 to both sides, then divide by 2.2.
2x1=72x-1=7
2x1+1=7+12x-1+1=7+1
2x=82x=8
2x2=82\dfrac{2x}{2}=\dfrac{8}{2}
x=4x=4
Now that xx has been isolated, we can see that the solution is x=4.x=4.
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Concept

Multi-Step Equations

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

Combining Like Terms:6=5x+2x8Variable Terms on Both Sides:x+1=-5x5Distributive Property:4(-6x+3)=12\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. 7y+2x3y2+x+5\begin{gathered} 7y+2x-3y-2+x+5 \end{gathered} This example expression contains three sets of like terms: x-terms,x\text{-terms}, y-terms,y\text{-terms}, and constants.\text{constants}.

Simplify algebraic expression 1 en.svg

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. +7y3y++2x+x+-2+54y +3x +3\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+2x8.6=5x+2x-8.

Solution
Notice there are two xx-terms on the right-hand side of the equation. To begin, we will combine these terms.
6=5x+2x86=5x+2x-8
6=7x86=7x-8
From here, we can solve the equation using inverse operations to isolate x.x. Specifically, we can add 88 to both sides, then divide by 7.7.
6=7x86=7x-8
6+8=7x8+86+8=7x-8+8
14=7x14=7x
147=7x7\dfrac{14}{7}=\dfrac{7x}{7}
2=x2=x
x=2x=2
The equation has the solution x=2.x=2.
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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.

3x=x23xx=xx22x=-2\begin{aligned} 3x&=x-2\\ 3x{\color{#0000FF}{-x}}&=x{\color{#0000FF}{-x}}-2\\ 2x&=\text{-} 2 \end{aligned}
Exercise

Solve x+1=-5x5.x+1=\text{-} 5x-5.

Solution
Notice in the equation that there are two variable terms — xx on the left-hand side and -5x\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 -5x\text{-} 5x to the left by adding 5x5x to the equation (the inverse operation of subtracting 55). After that, we can combine like terms.
x+1=-5x5x+1=\text{-} 5x-5
x+1+5x=-5x5+5xx+1+5x=\text{-} 5x-5+5x
6x+1=-56x+1=\text{-} 5
From here, we can isolate xx by subtracting 11 then dividing by 66 on both sides.
6x+1=-56x+1=\text{-} 5
6x+11=-516x+1-1=\text{-} 5-1
6x=-66x=\text{-} 6
6x6=-66\dfrac{6x}{6}=\dfrac{\text{-} 6}{6}
x=-1x=\text{-} 1
The equation has the solution x=-1.x=\text{-} 1.
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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.

Law of distr 1a.svg
Exercise

Solve 4(-6x+3)=12.4(\text{-} 6x+3)=12.

Solution
To solve the equation for x,x, we must first distribute 44 into the parentheses.
4(-6x+3)=124(\text{-} 6x+3)=12
4(-6x)+43=124(\text{-} 6x)+4\cdot 3=12
-24x+12=12\text{-} 24x+12=12
From here, we can subtract 1212 then divide by -24\text{-} 24 to isolate x.x.
-24x+12=12\text{-} 24x+12=12
-24x+1212=1212\text{-} 24x+12-12=12-12
-24x=0\text{-} 24x=0
-24x-24=0-24\dfrac{\text{-} 24x}{\text{-} 24}=\dfrac{0}{\text{-} 24}
x=0x=0
The equation has the solution x=0.x=0.
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Exercises

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