Problem 17 - College Algebra and Calculus: An Applied Approach

Question

In Exercises 17-54, solve the equation and check your solution. (Some equations have no solution.) $$ x+10=15 $$

Step-by-Step Solution

Answer

The solution to the equation \(x + 10 = 15\) is \(x = 5\).

Step 1: Identify the Unknown

Here, 'x' is the unknown that needs to be solved.

Step 2: Solving the Equation

Subtract 10 from both sides of the equation: \(x + 10 - 10 = 15 - 10\). This simplifies to \(x = 5\).

Step 3: Check the solution

Substitute \(x = 5\) into the original equation \(x + 10 = 15\). It simplifies to \(5 + 10 = 15\), which is a true statement. Therefore, the solution for \(x\) in the equation is correct.

Key Concepts

Algebraic Manipulation

Algebraic manipulation is a crucial skill in solving linear equations. It involves applying mathematical operations to both sides of an equation in order to isolate the unknown variable. In this exercise, our goal is to find the value of \(x\) by rearranging the equation \(x + 10 = 15\).
The key rule of algebraic manipulation is to perform the same operation on both sides of the equation. This keeps the equation balanced. Here, we need to get \(x\) by itself.

• Start by noticing that there is an additional \(10\) added to \(x\).
• To eliminate the \(10\), subtract \(10\) from both sides.
• This gives us \(x + 10 - 10 = 15 - 10\). The \(+10\) and \(-10\) cancel each other out.

As a result, we simplify the equation to \(x = 5\). Now, \(x\) is isolated and we have found its value.

Solving Equations

Solving linear equations is the process of finding a numerical value for a variable that makes the equation true. This involves both understanding the equation and strategically applying mathematical operations.
In the given exercise, the equation is \(x + 10 = 15\). To solve it, we used a simple yet effective method:

• Recognize that the equation involves adding \(10\) to \(x\).
• Perform the inverse operation to isolate \(x\). Here, that involved subtracting \(10\) on both sides to maintain balance.

This step-by-step approach leads us to the solution \(x = 5\). By solving the equation logically and systematically, we ensure our solution is valid.

Mathematical Reasoning

Mathematical reasoning plays a vital role in verifying solutions to equations. Once an equation is solved, we need to check that the solution truly satisfies the original equation. This involves substituting the found value back into the equation to ensure its validity.
In our scenario, we determined that \(x = 5\). To confirm this:

• Substitute \(x = 5\) back into the original equation \(x + 10 = 15\).
• Calculate it: \(5 + 10 = 15\).
• This results in a true statement, confirming our solution.

Checking your solutions reinforces the accuracy of your mathematical reasoning and ensures there are no computational mistakes. This insightful approach is essential in building confidence and precision in solving linear equations.