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One-Variable Equations

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.
2x1=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.
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Exercise

Solve 2x1=7.

Show Solution
Solution
To solve the equation, we will add 1 to both sides, then divide by 2.
2x1=7
2x1+1=7+1
2x=8
x=4
Now that x has been isolated, we can see that the solution is x=4.

Concept

Multi-Step Equations

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

Concept

Combining Like Terms

When simplifying algebraic expressions, it is possible to combine — add or subtractlike terms.
This example expression contains three sets of like terms, namely, x-terms, y-terms, and 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.
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Exercise

Solve 6=5x+2x8.

Show Solution
Solution
Notice there are two x-terms on the right-hand side of the equation. To begin, we will combine these terms.
6=5x+2x8
6=7x8
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=7x8
6+8=7x8+8
14=7x
2=x
x=2
The equation has the solution x=2.

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.

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Exercise

Solve x+1=-5x5.

Show Solution
Solution
Notice in the equation that there are two variable terms — x on the left-hand side and -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 to the left by adding 5x to the equation (the inverse operation of subtracting 5). After that, we can combine like terms.
x+1=-5x5
x+1+5x=-5x5+5x
6x+1=-5
From here, we can isolate x by subtracting 1 then dividing by 6 on both sides.
6x+1=-5
6x+11=-51
6x=-6
x=-1
The equation has the solution x=-1.

Rule

Distributive Property

Multiplying a number by the sum of two or more addends produces the same result as multiplying the number by each addend individually and then adding all the products together.

Note that the factor outside the parentheses is multiplied, or distributed, to every term inside. The Distributive Property is used to simplify expressions with parentheses.
a(b+c)=ab + ac
Since the Distributive Property is an axiom, it does not need a proof.
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Exercise

Solve 4(-6x+3)=12.

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