Problem 58 - Elementary Algebra

Question

For the following problems, simplify each of the algebraic expressions. $$ (a+5) 4+6 a-20 $$

Step-by-Step Solution

Answer

Question: Simplify the algebraic expression: (a + 5)4 + 6a - 20 Answer: 10a

Step 1: Distribute the 4 to both terms within the parentheses

Distribute the 4 to both (a) and (5) in the expression: \((a+5) 4 = 4a + 20\) The updated expression becomes: $$ 4a + 20 + 6a - 20 $$

Step 2: Combine like terms

Combine the terms with 'a' and the constant terms: $$ (4a + 6a) + (20 - 20) $$ 4a + 6a equals 10a, and 20 - 20 equals 0. So, the simplified expression is: $$ 10a + 0 $$ However, we can simplify it further as the "+0" has no effect on the value of the expression.

Step 3: Final Simplified Expression

The final simplified expression is: $$ 10a $$

Key Concepts

Simplification

Simplification is the process of reducing an algebraic expression to its simplest form. In this exercise, we began with the expression \((a+5) 4 + 6a - 20\). The goal was to simplify it into a form that is easier to manage.

Simplification often involves a few key steps:

• Applying mathematical properties like the distributive property
• Combining like terms
• Reducing any unnecessary constants or excess terms

In this case, we see that applying these steps helps in achieving a more concise and clearer expression of \(10a\). Each aspect of simplification brings the expression closer to its most efficient form.

Distributive Property

The distributive property is a fundamental algebraic tool that helps in simplifying expressions. It states that multiplying a sum by a number gives the same result as multiplying each addend individually by the number and then adding the products.

In our exercise, we applied the distributive property to the expression \((a+5) 4\). This means we distributed the 4 across both terms inside the parentheses.

• First multiply \(4\) by \(a\), resulting in \(4a\).
• Then multiply \(4\) by \(5\), resulting in \(20\).

The usage of the distributive property allows for a structured way to simplify parts of the expression, leading to an updated expression of \(4a + 20 + 6a - 20\). Understanding this property is key for tackling more complex algebraic problems.

Like Terms

Like terms are terms that have the same variable and can be combined. In algebra, combining like terms is a fundamental skill for simplifying expressions.

For the initial expression \(4a + 20 + 6a - 20\), the like terms are those with the variable 'a' and the constant terms.

• 4a and 6a: Both have the variable 'a'. They are combined to make \(10a\).
• 20 and -20: These are plain numbers (constants) and when summed, result in \(0\).

Combining like terms reduces the complexity of an expression, helping you arrive at \(10a\) as the simplest form. Recognizing and combining like terms is crucial for working efficiently with algebraic expressions.

Constants

Constants are numbers that stand alone in an expression and do not change; they do not contain any variables. They are the numerical counterpart in an equation, representing fixed values.

In our exercise, the constants were \(20\) and \(-20\). These appeared as part of the expression \(4a + 20 + 6a - 20\).

• When constants are added or subtracted, their values are simplified as part of the overall simplification process.
• In this instance, the addition of 20 and subtraction of 20 results in \(0\).

Constants are straightforward but essential components of algebra, allowing you to simplify expressions by combining and reducing parts of the expression that do not include variables.