Simplifying Expressions

In algebra, we often encounter expressions that can be complex or long. To make such expressions easier to work with, we simplify them. Simplifying an expression means making it as simple as possible.

What is the expression?

Expressions in algebra are combinations of numbers, variables, and mathematical operations (such as addition, subtraction, multiplication, and division).

3x + 2y - 5

Here, 3x + 2y - 5 is an expression where:

• 3x and 2y are terms having variables x and y.
• -5 is a constant term.

What is the meaning of simplification?

When we simplify an expression, our aim is to combine similar terms and make the expression shorter and more understandable. It is like cleaning a dirty table - you put similar items together and get rid of unnecessary ones.

Similar terms

Like terms are terms in an expression that have variables raised to the same power. You can only add like terms.

For example:

3x + 4x

Here, 3x and 4x are like terms because they both contain the variable x. You can add them:

3x + 4x = 7x

Steps to simplify expressions

1. Identify like terms: Look for terms that have the same variables.
2. Combine like terms: Add or subtract the coefficients of like terms.
3. Simplify constants: Perform operations on constant numbers.

Examples of simplifying expressions

Example 1

Simplify the expression by combining like terms.

5a + 3b + 2a - b

Solution:

First, identify like terms:

• 5a and 2a are like terms.
• 3b and -b are like terms.

Combine like terms:

• 5a + 2a = 7a
• 3b - b = 2b

The simplified expression is:

7a + 2b

Example 2

Simplify the expression.

4x + 7 + 3x - 5

Solution:

Identify and combine like terms:

• 4x + 3x = 7x
• 7 - 5 = 2

The simplified expression is:

7x + 2

Example 3: View

Simplify the expression.

2w + 3w + 4 - 2

Solution:

Combine like terms 2w and 3w to get 5w, and combine constants 4 and -2 to get 2.

The simplified expression is:

5w + 2

Distributive property

The distributive property helps simplify expressions that include parentheses. It states that multiplying a sum or difference by a number is the same as multiplying each addend separately and then adding the results.

a(b + c) = ab + ac

Let's use the distributive property to simplify an expression.

Example 4

Simplify the expression using the distributive property.

3(x + 4)

Solution:

Apply the distributive property:

• 3(x) = 3x
• 3(4) = 12

The simplified expression is:

3x + 12

Combination of multiple steps

Often, simplifying expressions may require a combination of methods, including combining like terms and using the distributive property.

Example 5

Simplify the expression.

2(3x + 4) + 5x - 2

Solution:

First, apply the distributive property:

2(3x + 4) = 6x + 8

The expression is as follows:

6x + 8 + 5x - 2

Then combine like terms 6x and 5x:

• 6x + 5x = 11x

Combine the constants 8 and -2:

• 8 - 2 = 6

The simplified expression is:

11x + 6

Practice problems

Try simplifying these expressions:

1. 7y + 2y - 3 + 8
2. 10m - 3 + 2(m + 5)
3. 4(2p - 1) + 5p
4. 6(a + 2) - a + 3

Check if you can apply the techniques discussed to simplify these expressions.

Conclusion

Simplifying expressions in algebra is a basic skill that makes working with mathematical expressions much easier. By combining like terms, using the distributive property, and performing arithmetic operations correctly, we can transform complex expressions into simpler, more useful forms. These strategies form the building blocks of more advanced algebra and are vital to solving equations effectively.

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