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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.Solve 2x−1=7.

Show Solution

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

$Combining Like Terms:Variable Terms on Both Sides:Distributive Property: 6=5x+2x−8x+1=-5x−54(-6x+3)=12 $

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.

Solve 6=5x+2x−8.

Show Solution

Notice there are two x-terms on the right-hand side of the equation. To begin, we will combine these terms.
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.
The equation has the solution x=2.

6=7x−8

AddEqn

LHS+8=RHS+8

6+8=7x−8+8

AddTerms

Add terms

14=7x

DivEqn

$LHS/7=RHS/7$

$714 =77x $

SimpQuot

Simplify quotient

2=x

RearrangeEqn

Rearrange equation

x=2

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.

Solve x+1=-5x−5.

Show 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.
From here, we can isolate x by subtracting 1 then dividing by 6 on both sides.
The equation has the solution x=-1.

6x+1=-5

SubEqn

LHS−1=RHS−1

6x+1−1=-5−1

SubTerms

Subtract terms

6x=-6

DivEqn

$LHS/6=RHS/6$

$66x =6-6 $

SimpQuot

Simplify quotient

x=-1

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.

Since the Distributive Property is an axiom, it does not need a proof.

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

Show Solution

To solve the equation for x, we must first distribute 4 into the parentheses.
From here, we can subtract 12 then divide by -24 to isolate x.
The equation has the solution x=0.

-24x+12=12

SubEqn

LHS−12=RHS−12

-24x+12−12=12−12

SubTerm

Subtract term

-24x=0

DivEqn

$LHS/-24=RHS/-24$

$-24-24x =-240 $

SimpQuot

Simplify quotient

x=0

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