Problem 51 - Elementary Algebra Within Reach

Question

In Exercises \(47-66\), simplify the expression by removing symbols of grouping and combining like terms. $$ m-3(m-7) $$

Step-by-Step Solution

Answer

The simplified expression is \( -2m + 21 \).

Step 1: Distribute the multiplication

Distribute \( -3 \) across the terms inside the parentheses. This becomes: \( m - 3m + 21 \)

Step 2: Combine like terms

Combine \( m \) and \( -3m \) terms together. This results in: \( -2m + 21 \)

Key Concepts

The Magic of the Distributive Property

The distributive property is a fundamental tool in algebra that helps us simplify expressions by eliminating parentheses. It states that if you have a term outside the parentheses, you can distribute it to each term inside the parentheses.
For example, consider the expression \( a(b + c) \). According to the distributive property, this can be expanded to \( ab + ac \).
This property is especially useful in breaking down more complex algebraic expressions, making them easier to work with.

• In our example, the expression is \( m - 3(m-7) \).
• We use the distributive property by multiplying \( -3 \) with each term inside the parentheses, which simplifies to \( m - 3m + 21 \).

Notice how we distributed the \( -3 \) and turned the subtraction into addition when multiplying with a negative number.

Combining Like Terms to Simplify Expressions

After using the distributive property, the next step in simplifying algebraic expressions is combining like terms. Like terms are terms that have the same variable parts, raised to the same power.
In simpler terms, these are terms that can be added or subtracted together because they share the same base and exponent.

• For instance, in our resulting expression \( m - 3m + 21 \), the like terms are \( m \) and \( -3m \).
• Combine these to simplify the expression to \( -2m + 21 \).

This process reduces the expression to fewer terms, making it easier to evaluate or plug into further equations.

Mastering Algebraic Manipulation

Algebraic manipulation is the art of rearranging and simplifying algebraic expressions using various algebraic rules and properties. This skill is essential for solving equations and for transforming expressions into more manageable forms.
The techniques we used in the previous sections, like the distributive property and combining like terms, are just a few examples of algebraic manipulation.

• Learning these skills allows for the simplification of complex expressions and aids communication in mathematical problem-solving.
• Think of algebraic manipulation as a toolkit; each tool helps in different scenarios, allowing you to approach problems efficiently and creatively.

By practicing these techniques, such as distributing and combining, you'll enhance your ability to work through intricate algebraic expressions confidently.