Introduction to Simplifying Expressions and Solving Equation Word Problems
Mathematics may be defined as the economy of counting. There is no problem in the whole of mathematics which cannot be solved by direct counting.
—ERNST MACH (1838–1916)
This lesson reviews the key words and phrases for the basic operations and provides examples and tips on simplifying algebraic expressions and the equation solving steps. Equation word problems are modeled with explanations to help your understanding in this type of question.
Key Words and Phrases
Translating expressions from words into mathematical symbols was covered in Lesson 1. The following chart below summarizes the key words and phrases studied in that lesson for the four basic operations and the equal sign.
Refer to this chart when you are changing sentences in words to math equations.
Tip:
In algebra, the number in front of the letter is called the coefficient and the letter is called the variable. In the expression 8x, 8 is the coefficient and x is the variable.
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Simplifying Expressions - Combining Like Terms and The Distributive Property
Two important processes to know when you are simplifying expressions are combining like terms and the distributive property.
Combining Like Terms
Terms, in mathematics, are numbers and symbols that are separated by addition and subtraction.
The expressions 3, 5x, and 7xy are each one term.
The expressions 2x + 3, and x – 7 each have two terms.
The expression 7x + 5y – 9 has three terms.
Like terms are terms with the same variable and exponent. Like terms can be combined by addition and subtraction. To do this, add or subtract the coefficients and keep the variable the same. For example 3x + 5x = 8x, and 6y2 – 4y2 = 2y2.
Tip:
Be sure to combine only like terms. Terms without the exact same variable and exponent cannot be combined: 5x2 and 6x cannot be combined because the exponents are not the same.
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Distributive Property
The distributive property is used when a value needs to be multiplied, or distributed, to more than one term. For example, in the expression 3(x + 10), the number 3 needs to be multiplied by the term x and the term 10. The use of arrows can help in this process, as shown in the following figure.
The result becomes 3 × x + 3 × 10, which simplifies to 3x + 30.
Solving Equations
When you are solving equations, the goal is to get the letter, or variable, by itself. This is called isolating the variable. Each of the following examples goes through the process of isolating the variable for different types of equations.
Tip:
One of the most important rules in equation solving is to do the same thing on both sides of the equation. For example, if you divide on one side to get the variable alone, divide the other side by the same number. This keeps the equation balanced and will lead to the correct solution.
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Solving Equations with Multiple Steps
One-Step Equations
One-step equations are named this because they have only one operation to undo, so they should take only one step to solve. Use inverse, or opposite, operations to get the variable alone. This is called isolating the variable.
Example: Solve the equation for x.
| 3x = 12 |
|
 |
Divide each side by 3, the inverse operation of multiplying by 3. |
| x = 4 |
The variable is alone. |
Two-Step Equations
Two-step equations have two operations, and therefore take two steps to solve.
Example: Solve the equation for x.
 – 10 = 10 |
|
| –10 + 10 = 10 + 10 |
Add 10 to each side, the inverse operation of subtracting 10. |
| = 20 |
Simplify. |
| × 4 = 20 × 4 |
Multiply each side by 4, the inverse operation of dividing by 4. |
| x = 80 |
The variable is alone. |
Variables on Both Sides of the Equation
When there are variables on both sides of the equation, get rid of the variable with the smaller coefficient by using the inverse operation.
Example: Solve the equation for x.
| 7x = 3x + 28 |
|
| 7x – 3x = 3x – 3x + 28 |
Subtract 3x from each side of the equation. |
| 4x = 28 |
Simplify. |
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Divide each side by 4, the inverse operation of multiplying by 4. |
| x = 7 |
The variable is alone. |
Multistep Equations
To solve multistep equations, you may first need to use the distributive property and combine like terms in order to simplify. If there are variables on both sides of the equation, handle them next. Finally, use the inverse operations to isolate the variable in the one- or two-step equation.
Example: Solve the equation for x.
| 2(x – 6) – x = 5x |
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| 2x – 12 – x = 5x |
Get rid of the parentheses by using the distributive property. |
| x – 12 = 5x |
Combine like terms. |
| x – x – 12 = 5x – x |
Subtract x from each side of the equation. |
| –12 = 4x |
Simplify. |
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Divide each side by 4, the inverse operation of multiplying by 4. |
| –3 = x |
The variable is alone. |
Tip:
The equation solving steps can be summarized as the following:
- Get rid of parentheses.
- Combine like terms.
- Handle variables of both sides of the equation.
- Solve the one- or two-step equation.
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Equation Word Problems
Now that the steps to solving equations have been practiced, let's apply these steps to solving word problems involving equations. Use the chart at the beginning of the lesson and the examples in Lesson 1 for help with translating phrases into math symbols and equations. Then, use your knowledge and skills in equation solving to find the correct solution to each problem. In addition, use the word-problem solving steps to be sure each detail is taken care of and all problems are checked.
Tip:
To check solutions in equations, substitute the value into the original equation and use order of operations. The correct order of operations is Parentheses, Exponents, Multiplication and Division, Addition and Subtraction. It is commonly remembered as the acronym PEMDAS.
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Example 1
Ten more than a number is equal to 40. What is the number?
Read and understand the question. This question is looking for a number when clues about this number are given.
Make a plan. Translate the statement into equation form. Then, solve the equation using the equation solving steps.
Carry out the plan. Let x = a number. The key phrase more than means addition. The statement translates to x + 10 = 40. Next, subtract 10 from each side of the equation to get the variable alone.
Check your answer. Check your solution by substituting the answer into the equation.
becomes
This answer is checking.
Example 2
Eight less than twice a number is equal to four times the number. What is the number?
Read and understand the question. This question is looking for a number when clues about this number are given.
Make a plan. Translate the statement into equation form. Then, solve the equation using the equation solving steps.
Carry out the plan. Let x = a number. The key phrase less than means subtraction and twice a number is written as 2x. The first part of the statement translates to 2x – 8. In the second part of the sentence, four times the number is written as 4x. The entire equation is
Get the variables on one side of the equation by subtracting 2x from each side.
The equation simplifies to
Next, divide each side by 2 to get the variable alone.
Check your answer. Check your solution by substituting the answer into the equation.
becomes
This answer is checking.
Example 3
Forty-two added to a number is equal to 6 times the sum of the number and 2. What is the number?
Read and understand the question. This question is looking for a number when clues about this number are given.
Make a plan. Translate the statement into equation form. Then, solve the equation using the equation solving steps.
Carry out the plan. Let x = a number. The key phrase added to means addition. The first part of the statement translates to x + 42. The second part of the sentence, six times the sum of a number and 2 is written as 6(x + 2). The entire equation is
Use the distributive property on the right side to make the equation
Get the variables on one side of the equation by subtracting x from each side.
The equation simplifies to
Subtract 12 from each side of the equation to get 30 = 5x. Next, divide each side by 5 to get the variable alone:
Check your answer. Check your solution by substituting the answer into the equation.
becomes
This answer is checking.
Find practice problems and solutions for these concepts at Simplifying Expressions and Solving Equation Word Problems Practice Questions.
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