Problem 41 - Algebra 1: Concepts and Skills

Question

Simplify the expression. $$ 4(3 a+5)+3(-4 a+2) $$

Step-by-Step Solution

Answer

The simplified form of the given expression \(4(3 a+5)+3(-4 a+2)\) is 26.

Step 1: Apply the Distributive Property

The distributive property states that \( a(b + c) = ab + ac \). So, apply this property to both terms to simplify: \( 4(3a + 5) + 3(-4a + 2) \ = 12a + 20 - 12a + 6 \)

Step 2: Combining like terms

Like terms are expressions with the same variable and exponent. Two like terms can be combined into one. After applying the distributive property, we can combine like terms. \( 12a - 12a + 20 + 6 \ = 0a + 26 \)

Step 3: Final Simplification

0 times any number is 0. So, the simplified form can disregard the term with the variable as its coefficient is zero. \( 0a + 26 \ = 26 \)

Key Concepts

Distributive Property

The distributive property is a useful tool in algebra that allows us to multiply a single term by each term inside a parenthesis. It is usually expressed as \(a(b + c) = ab + ac\). This means that if you have an expression like \(4(3a + 5)\), you simplify it by multiplying the 4 by each of the terms inside the parentheses, which will give you \(4\times 3a\) and \(4\times 5\). Following the distributive property, our expression becomes \(12a + 20\).

Using the distributive property makes it easier to work with complex expressions, especially when combining like terms, which is the next step in simplifying an algebraic expression. Understanding this property is crucial because it is frequently used not only in algebra but in higher-level mathematics as well.

Combining Like Terms

Combining like terms is a method used to simplify algebraic expressions by merging terms that have the same variable raised to the same power. For instance, \(12a\) and \(\-12a\) are like terms because they both contain the variable \(a\) to the first power. When you combine \(12a\) and \(\-12a\), they effectively cancel each other out because \(12a - 12a = 0a\).

This simplification process makes the expression easier to work with or solve. After applying the distributive property, it's important to look for and combine like terms to make sure the expression is simplified as much as possible. The more complex the expression, the more you'll appreciate the power of combining like terms to make your calculations easier.

Eliminating Zero Terms

Eliminating zero terms involves recognizing when parts of an algebraic expression equate to zero and can therefore be removed to simplify the expression. Since anything multiplied by zero is zero, any term in the expression that has a coefficient of zero can be disregarded. For example, in the expression \(0a + 26\), \(0a\) is a zero term and simplifies to \(0\), leaving us with just \(26\).

Removing zero terms is often the final step in simplification. By eliminating them, we clean up our expression, making it far easier for us to understand and further manipulate, if necessary. It's essential to remember that a zero coefficient renders the entire term irrelevant in terms of value to the overall expression.