Problem 7 - Precalculus: Mathematics for Calculus

Question

Yes or No? If No , give a reason. Assume that \(a\) and \(b\) are nonzero real numbers. (a) Is \(a-b\) equal to \(b-a ?\) (b) Is \(-2(a-5)\) equal to \(-2 a-10 ?\)

Step-by-Step Solution

Answer

(a) No, subtraction is not commutative. (b) No, the expressions are not equivalent after simplification.

Step 1: Analyzing Part (a)

The expression we need to analyze is whether \(a - b\) is equal to \(b - a\). This question is asking if two expressions are equivalent.

Step 2: Understanding Part (a): Mathematical Operation of Subtraction

Subtraction is not commutative, meaning \(a - b\) is not generally equal to \(b - a\). In fact, \(b - a\) is equal to \(-1\times(a - b)\).

Step 3: Conclusion for Part (a)

Since \(a - b\) is equal to the negative of \(b - a\) and not equal to \(b - a\), the answer is "No."

Step 4: Analyzing Part (b)

Examine if the expression \(-2(a-5)\) is equal to \(-2a - 10\). This requires us to simplify \(-2(a-5)\) to check if it equals the right side.

Step 5: Expanding the Expression in Part (b)

Apply the distributive property: Multiply \(-2\) with both terms inside the parenthesis: \(-2(a-5) = -2 \cdot a + -2 \cdot (-5) = -2a + 10\).

Step 6: Comparison in Part (b)

Compare the expanded form \(-2a + 10\) with the expression \(-2a - 10\). They have different constant terms (+10 vs -10), meaning they are not equal.

Step 7: Conclusion for Part (b)

Since \(-2(a-5)\) simplifies to \(-2a + 10\) and not \(-2a - 10\), the answer is "No."

Key Concepts

Subtraction

Subtraction is a fundamental arithmetic operation and involves finding the difference between numbers or algebraic expressions. It's like removing or taking away quantities. Unlike addition and multiplication, subtraction is not commutative. This means that the order in which you subtract numbers matters a lot. For example, if you have two numbers, say 5 and 3, then:

• Subtracting 3 from 5, written as: \(5 - 3\), gives 2.
• But subtracting 5 from 3, expressed as \(3 - 5\), results in -2.

This shows that \(a - b\) does not equal \(b - a\). In fact, the relation between them is \(b - a = -1 \times (a - b)\). This is key in algebra because swapping operands transforms the expression significantly. So, always be cautious about the sequence in subtraction, especially when working with variables such as in the original exercise.

Distributive Property

The distributive property is an essential algebraic rule that connects multiplication and addition or subtraction. It allows us to "distribute" a single factor across terms inside a parenthesis.If you have an expression like \(-2(a-5)\), the distributive property lets you multiply each term inside the parenthesis by \(-2\). Here's how it works step-by-step:

• Multiply \(-2\) times \(a\) to get \(-2a\).
• Multiply \(-2\) times \(-5\) to get \(+10\).
• Combine these results to get \(-2a + 10\).

This property is vital for simplifying expressions and solving equations. It ensures that you correctly distribute multiplication over each term inside the brackets, whether in addition or subtraction cases. Keep in mind that neglecting this can lead to wrong conclusions. The initial misunderstanding in the exercise highlights the power of the distributive property.

Commutative Property

The commutative property is a principle that applies to addition and multiplication, but not to subtraction or division. It states that changing the order of the operands doesn't change the result. For instance:- In addition, \(a + b = b + a\).- In multiplication, \(a \times b = b \times a\).However, subtraction fails to follow this property. Let's revisit why using subtraction is different:

• For subtraction, \(a - b\) does not equal \(b - a\). This difference highlights how subtraction needs careful attention to order.
• In the algebraic context, swapping terms can change the sign, precisely reflecting how the operation diverges from commutative cases.

Understanding the boundaries of the commutative property helps prevent errors in algebraic manipulations. It's crucial to remember which operations respect this property and adjust practices accordingly as seen in the exercise.