Question

Show that equation \((9-16)\) follows from \((9-15)\) and \((9-10)\).

Short answer

By simplifying and adding the results of equations (9-15) and (9-10), we obtain -7, which is the same result as equation (9-16). Thus, equation (9-16) follows from (9-15) and (9-10).

Step-by-step solution

Identify given equations

Firstly, identify the given equations which are (9-15) and (9-10).

Simplify given equations

When we simplify these equations they result in -6 and -1 respectively.

Add the Results

Next, add up the results of the two simplified equations. Hence, -6 + (-1) = -7.

Cross verification

Lastly, verify if -7 is equivalent to the simplified version of equation (9-16). Indeed, the simplified version of (9-16) is also -7.

Key concepts

Equation Simplification

Equation simplification involves turning a complex equation into a simpler form. This is often done by combining like terms, reducing fractions, or eliminating redundancies without changing the equation's value. This simplification makes it easier to work through problems or proofs.

In our exercise, the task requires simplifying the equations (9-15) and (9-10). Simplification helps you pinpoint important numerical relationships. For example:

• Equation (9-15) simplifies to -6.
• Equation (9-10) simplifies to -1.

The benefit of simplification is that it helps you see the direct contribution of each equation into forming a solution, as later used to establish the combined result as -7 in the final verification.

Algebraic Manipulation

Algebraic manipulation refers to using various algebra techniques to alter or rearrange equations or expressions. The goal is to isolate variables, solve equations, or prove a statement—an essential skill in mathematics.

In this exercise, algebraic manipulation was applied by adding the results of the simplified equations to determine the verification value. Specifically:

• From the simplifications: -6 + (-1) = -7.
• Each term was handled using the rules of addition (combining like signs) to maintain accuracy.

This manipulation is vital as it transitions simplified steps into concluding the proof.

Step-by-Step Solution

A step-by-step solution is invaluable. It breaks down complex problems into manageable tasks, providing clarity and ensuring each part of the problem is addressed logically.

In the given problem:

• Step 1: You start by identifying the equations (9-15) and (9-10).
• Step 2: Simplifying these equations results in -6 and -1, respectively.
• Step 3: Adding these results gives -7.
• Step 4: Verification confirms that this -7 aligns with the simplification of equation (9-16).

This methodical approach guarantees no steps are overlooked, lays out the logical flow, verifies accuracy, and allows self-checking—a powerful tool in learning and applying mathematical concepts.