Problem 57 - Elementary Algebra

Question

Use the commutative properties of addition and multiplication to write equivalent expressions for the following problems. $$ a+4 b $$

Step-by-Step Solution

Answer

Answer: Some equivalent expressions for the given problem are \(4b+a\), \(a+b\cdot4\), and \(b\cdot4+a\).

Step 1: Identify commutative property of addition

The given expression is \(a+4b\). We can use the commutative property of addition to rewrite this expression in a different order. According to the commutative property, \(a + b = b + a\). Therefore, we can rewrite the given expression as \(4b + a\).

Step 2: Identify commutative property of multiplication

Now we can look for the commutative property of multiplication within the expression \(4b\). According to the commutative property, \(ab = ba\). Therefore, we can rewrite \(4b\) as \(b \cdot 4\).

Step 3: Combine the commutative properties

Now we combine the commutative properties of addition and multiplication. We can rewrite the original expression, \(a+4b\), using these equivalent expressions as follows: - \(4b+a\) - \(a+b\cdot4\) - \(b\cdot4+a\) These are all equivalent expressions for the original problem, \(a+4b\), using the commutative properties of addition and multiplication.

Key Concepts

Commutative Property of Addition

The Commutative Property of Addition is a fundamental principle in algebra that defines how numbers can be added in any order without changing the result. For instance, when you have two numbers, like 3 and 5, whether you add 3 to 5 or 5 to 3, the sum will always be 8.

Mathematically, this property is written as: \(a + b = b + a\). Let’s consider the example with variables like in the exercise \(a+4b\). Using this property, we can rearrange the terms to get \(4b + a\). As long as the operation is addition, the order of numbers doesn't matter. This concept is crucial not just in complex problems but in our daily mathematical understanding as well.

Commutative Property of Multiplication

Similar to addition, the Commutative Property of Multiplication allows us to multiply numbers in any order. This makes it easier when working with numbers as you can arrange them as needed for convenience or clarity.

Expressed in algebraic terms, it means \(ab = ba\). In the context of our exercise, we look at the term \(4b\). By applying this property, we can write it as \(b \cdot 4\). This is particularly useful for mental math or when working with variables and coefficients since it allows more flexibility in the arrangement of terms for simplification or further operations.

Equivalent Expressions

Equivalent expressions are different expressions that have the same value, regardless of what values are substituted for the variables they contain. They are a cornerstone in understanding algebraic relationships.

In our example, the expressions \(4b + a\), \(a + b \cdot 4\), and \(b \cdot 4 + a\) are all equivalent. They might look different, but they represent the same quantity. It's like having different recipes for the same dish – the ingredients and steps might vary, but the end result is the same delicious meal. This concept aids in simplifying expressions and solving equations efficiently.

Elementary Algebra

Elementary Algebra is the branch of mathematics that deals with variables, expressions, and equations. It serves as the foundation for more advanced topics within algebra and beyond.

In this field, principles like the Commutative Properties of Addition and Multiplication play a crucial role in understanding how expressions can be manipulated and equivalent expressions can be formed. Whether working with simple number operations or complex expressions, the ability to rearrange and simplify thanks to these properties is essential. In our textbook problem, we applied these basic algebraic properties to reorder the expression \(a+4b\), showcasing how fundamental concepts of elementary algebra are applied.