Problem 39 - Algebra 2

Question

Simplify each expression. $$ 2(10 m-7 a)+3(8 a-3 m) $$

Step-by-Step Solution

Answer

The simplified expression is \( 11m + 10a \).

Step 1: Distribute the coefficients across the terms in the first bracket

First, distribute the 2 across the terms inside the first parenthesis. This gives us: \( 2 \times 10m - 2 \times 7a = 20m - 14a \).

Step 2: Distribute the coefficients across the terms in the second bracket

Now, distribute the 3 across the terms inside the second parenthesis. This gives us: \( 3 \times 8a - 3 \times 3m = 24a - 9m \).

Step 3: Combine like terms

Now, combine the like terms from both distributed expressions. Combine the \( m \) terms: \( 20m - 9m = 11m \). Combine the \( a \) terms: \( -14a + 24a = 10a \).

Step 4: Write out the simplified expression

The simplified expression combines the results of the like terms as \( 11m + 10a \).

Key Concepts

Distributive Property

The Distributive Property is an important algebraic rule. It helps us multiply a single term by two or more terms inside parentheses.
This process "distributes" the single term across all the terms inside the parentheses. For example, in the expression \( 2(10m - 7a) \), we apply the distributive property as follows:

• Multiply 2 by the first term inside the parentheses, which is \( 10m \). We get \( 2 \times 10m = 20m \).
• Next, multiply 2 by the second term \(-7a\), resulting in \( 2 \times -7a = -14a \).

Repeating this with another set of terms, \( 3(8a - 3m) \), uses the same principle:

• Multiply 3 by \( 8a \) to get \( 24a \).
• Multiply 3 by \(-3m\) for \(-9m\).

Remember, distribute carefully, ensuring that each term inside the brackets is multiplied by the number outside.

Combining Like Terms

After using the distributive property, you often end up with terms that can be combined. "Like terms" are terms that have exactly the same variable components. In our example, we need to combine the terms obtained after distribution: \( 20m - 14a \) and \( 24a - 9m \).

• For "\( m \)" terms: Combine \( 20m \) and \(-9m\). Add the coefficients (the numbers in front of the variables) to get \( 11m \).
• For "\( a \)" terms: Combine \(-14a\) and \( 24a \). Add these as well to get \( 10a \).

Combining like terms simplifies the expression by reducing it to fewer terms. It makes it easier to understand or use it in further calculations. Practice combining to know quickly if terms can be added together.

Simplification of Expressions

Simplifying expressions is the process of making an expression as simple as possible. It involves applying the distributive property and combining like terms, as demonstrated. Once these steps are completed, you can write the expression in its simplest form. With the example we've been working on, we've already passed these stages to get to \( 11m + 10a \).

• This final expression is as concise as possible because no further simplification can be done.
• Simplifying helps both in solving algebraic equations and in understanding their relationships more clearly.

Regular practice helps you identify which expressions can be simplified further. Plus, it boosts confidence in mathematical operations, paving the way for more advanced algebraic concepts.