2.1 Solve equations using the subtraction and addition properties  (Page 3/6)

You should always simplify as much as possible before you try to isolate the variable. Remember that to simplify an expression means to do all the operations in the expression. Simplify one side of the equation at a time. Note that simplification is different from the process used to solve an equation in which we apply an operation to both sides.

How to solve equations that require simplification

Solve: $9x-5-8x-6=7.$

Solution

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Solve: $5(n-4)-4n=\mathrm{-8}.$

Solution

We simplify both sides of the equation as much as possible before we try to isolate the variable.

Distribute on the left.
Use the Commutative Property to rearrange terms.
Combine like terms.
Each side is as simplified as possible. Next, isolate $n$ .
Undo subtraction by using the Addition Property of Equality.
Add.
Check. Substitute $n=12$ .
	The solution to $5(n-4)-4n=\mathrm{-8}$ is $n=12.$

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Solve: $3(2y-1)-5y=2\left(y+1\right)-2\left(y+3\right).$

Solution

We simplify both sides of the equation before we isolate the variable.

Distribute on both sides.
Use the Commutative Property of Addition.
Combine like terms.
Each side is as simplified as possible. Next, isolate $y$ .
Undo subtraction by using the Addition Property of Equality.
Add.
Check. Let $y=\mathrm{-1}$ .
	The solution to $3(2y-1)-5y=2\left(y+1\right)-2\left(y+3\right)$ is $y=\mathrm{-1}.$

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Translate to an equation and solve

To solve applications algebraically, we will begin by translating from English sentences into equations. Our first step is to look for the word (or words) that would translate to the equals sign . [link] shows us some of the words that are commonly used.

The steps we use to translate a sentence into an equation are listed below.

Translate and solve: Eleven more than x is equal to 54.

Solution

Translate.
Subtract 11 from both sides.
Simplify.
Check: Is 54 eleven more than 43? $\begin{array}{ccc}\hfill 43+11& \stackrel{?}{=}\hfill & 54\hfill \\ \hfill 54& =\hfill & 54✓\hfill \end{array}$

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Translate and solve: The difference of $12t$ and $11t$ is $\mathrm{-14}$ .

Solution

Translate.
Simplify.
Check: $\begin{array}{ccc}\hfill 12(\mathrm{-14})-11(\mathrm{-14})& \stackrel{?}{=}\hfill & \mathrm{-14}\hfill \\ \hfill \mathrm{-168}+154& \stackrel{?}{=}\hfill & \mathrm{-14}\hfill \\ \hfill \mathrm{-14}& =\hfill & \mathrm{-14}✓\hfill \end{array}$

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Translate and solve applications

Most of the time a question that requires an algebraic solution comes out of a real life question. To begin with that question is asked in English (or the language of the person asking) and not in math symbols. Because of this, it is an important skill to be able to translate an everyday situation into algebraic language.

We will start by restating the problem in just one sentence, assign a variable, and then translate the sentence into an equation to solve. When assigning a variable, choose a letter that reminds you of what you are looking for. For example, you might use q for the number of quarters if you were solving a problem about coins.