Problem 29 - Algebra A Combined Function

Question

Use the commutative and associative properties to simplify each expression. See Examples 5 and 6. $$ -\frac{1}{2}(5 x) $$

Step-by-Step Solution

Answer

The simplified expression is \(-\frac{5}{2}x\).

Step 1: Identify the expression

The given expression to simplify is \(-\frac{1}{2}(5x)\) using the commutative and associative properties.

Step 2: Use the Commutative Property

The Commutative Property says that you can change the order of multiplication without affecting the result. Therefore, you can rewrite the expression as: \(-\frac{1}{2} \times 5 \times x\)

Step 3: Use the Associative Property

The Associative Property states you can regroup the numbers as you like when multiplying. Group \(-\frac{1}{2}\) and 5: \((-\frac{1}{2} \times 5) \times x\).Simplify the multiplication inside the parentheses.

Step 4: Simplify the multiplication

Calculate the product inside the parentheses: \(-\frac{1}{2} \times 5 = -\frac{5}{2}\). This gives you the expression \(-\frac{5}{2}x\).

Step 5: Conclusion

The simplified form of the given expression \(-\frac{1}{2}(5x)\) is \(-\frac{5}{2}x\).

Key Concepts

Commutative Property

The Commutative Property is a fundamental principle in algebra that can make your calculations more flexible and easier to handle. Simply put, this property states that in addition and multiplication, you can change the order of the numbers without affecting the result. In mathematical terms, this means:

• For addition: \( a + b = b + a \)
• For multiplication: \( a \cdot b = b \cdot a \)

In the context of the exercise, we're dealing with multiplication, specifically with the expression \(-\frac{1}{2}(5x)\). Here, the commutative property allows us to rearrange the multiplying factors:

• Originally: \(-\frac{1}{2} \cdot (5 \cdot x)\)
• Rearranged: \(-\frac{1}{2} \cdot 5 \cdot x\)

By using the commutative property, we position the numbers in a way that makes further calculations, like those using the associative property, simpler and more straightforward.

Associative Property

The Associative Property in algebra is incredibly useful when dealing with more complex expressions because it allows you to change the grouping of numbers without affecting the result. For example, if you have three or more numbers being multiplied, this property lets you decide which numbers to multiply first. The general rule can be expressed as:

• For addition: \( (a + b) + c = a + (b + c) \)
• For multiplication: \( (a \cdot b) \cdot c = a \cdot (b \cdot c) \)

In the exercise provided, once we have the expression \(-\frac{1}{2} \times 5 \times x\), the associative property allows us to group \(-\frac{1}{2}\) and \(5\) together:

• Change grouping: \((-\frac{1}{2} \cdot 5) \cdot x\)

This simplifies the expression more easily because you can focus on solving smaller parts at a time. Simplifying \(-\frac{1}{2} \cdot 5\) first, you get \(-\frac{5}{2}\), which drastically simplifies the task of working with larger algebraic expressions.

Multiplication of Fractions

Multiplying fractions is a key skill, and understanding it can make working with algebraic expressions significantly more manageable. When multiplying fractions, the general rule is to multiply the numerators and multiply the denominators separately, then simplify if possible:

• If \(\frac{a}{b} \cdot \frac{c}{d}\), then it equals to \(\frac{a \cdot c}{b \cdot d}\).

In the exercise, you initially have the expression \((-\frac{1}{2} \times 5) \cdot x\). Focusing on multiplying the fractions, \(-\frac{1}{2}\) and \(5\) (which can be considered as \(\frac{5}{1}\)):

• Numerator: \(-1 \times 5 = -5\)
• Denominator: \(2 \times 1 = 2\)

Thus, transforming this into \(-\frac{5}{2}\). This fraction multiplication simplifies our expression effectively as it results in \(-\frac{5}{2}x\). Mastering this concept shortens the path to finding solutions and makes algebraic tasks less daunting.