Problem 25 - Algebra and Trigonometry with Analytic Geometry

Question

\(-(\mathbf{a}+\mathbf{b})=-\mathbf{a}-\mathbf{b} \quad\)

Step-by-Step Solution

Answer

The equality holds true after distributing the negative sign and simplifying.

Step 1: Distribute the Negative Sign

We start by distributing the negative sign across the parentheses in the expression \[-(\mathbf{a}+\mathbf{b})\]When you distribute a negative sign, you apply it to each term inside the parentheses:\[-(\mathbf{a}) + (-\mathbf{b})\]

Step 2: Simplify the Expression

The expression we obtained in the previous step is:\[-\mathbf{a} + (-\mathbf{b})\]This simplifies to:\[-\mathbf{a} - \mathbf{b}\]

Step 3: Verify Equality

Now, compare the simplified expression:\[-\mathbf{a} - \mathbf{b}\]with the right side of the given equation:\[-\mathbf{a} - \mathbf{b}\]These are identical, hence the equality is verified.

Key Concepts

Negative Sign Distribution

In algebra, distributing a negative sign is essential in simplifying expressions correctly. When you come across a negative sign outside a set of parentheses, like in \( -( extbf{a} + extbf{b}) \), you need to "distribute" this negative sign to each term inside the parentheses.
This means that the negative sign changes the sign of each term inside, similar to multiplying by \(-1\).

• For the term \( extbf{a} \), it turns into \( - extbf{a} \).
• For the term \( extbf{b} \), it turns into \( - extbf{b} \).

So, when distributing the negative sign in \( -( extbf{a} + extbf{b}) \), the expression becomes \( - extbf{a} - extbf{b} \). Understanding this concept will help prevent mistakes and simplify your calculations accurately.

Simplifying Algebraic Expressions

Simplifying algebraic expressions involves performing operations to present the expression in its simplest form.
After you've distributed all terms, the goal is to combine like terms if possible and present the expression neatly.
In our case, we had \[- extbf{a} + (- extbf{b})\] post-distribution.The expression \(- extbf{a} + (- extbf{b})\) simplifies to \(- extbf{a} - extbf{b}\).
This is because adding a negative number is the same as subtracting that number.

• Always ensure you've distributed any constants or negative signs properly before simplifying any terms.
• Look for opportunities to combine similar terms for the cleanest expression.

Effective simplification not only makes the expression easier to work with, but it also helps in understanding broader algebra problems.

Verification of Equations

The final step in dealing with algebraic expressions like these is verifying the equation.
Verification means checking whether both sides of an equation are equal after you have simplified.To verify, compare the left-hand side of the equation with the right-hand side. For \(-( extbf{a} + extbf{b}) = - extbf{a} - extbf{b}\), we simplified the left side to \(- extbf{a} - extbf{b}\), which matches perfectly with the right side.
This matching confirms the equation holds true.

• A correct verification implies you've done all prior steps, like distribution and simplification, correctly.
• Verification builds confidence that the solution is accurate and helps catch mistakes early in the solving process.

Taking this step ensures that you understand the solution fully and that you've maintained equation balance throughout.