Problem 2 - Algebra A Combined Function

Question

Solve each equation. Check each solution. $$ x+14=25 $$

Step-by-Step Solution

Answer

The solution is \( x = 11 \).

Step 1: Write down the equation

The given equation to solve is \( x + 14 = 25 \).

Step 2: Isolate the variable

To solve for \( x \), we need to isolate it. We do this by subtracting 14 from both sides of the equation. So the operation is \( x + 14 - 14 = 25 - 14 \).

Step 3: Simplify both sides

After performing the subtraction, the left side simplifies to \( x \) and the right side simplifies to 11. Hence, \( x = 11 \).

Step 4: Check the solution

Replace \( x \) with 11 in the original equation to verify the solution: \( 11 + 14 = 25 \). Calculating the left side gives 25, which equals the right side, confirming \( x = 11 \) is correct.

Key Concepts

Solving Equations

Solving equations is about finding the value of the variable that makes the equation true. An equation is like a balance scale; whatever you do to one side, you must do to the other. Let's take the equation given: \( x + 14 = 25 \).

• To isolate the variable \( x \), we perform operations that get \( x \) by itself on one side of the equation. Here, we need to "get rid" of the \(+ 14\).
• The opposite operation of addition is subtraction.

So, we subtract 14 from both sides:\[ x + 14 - 14 = 25 - 14 \]This operation keeps the balance of the equation while removing the 14 from the left side, ultimately helping us solve for \( x \). Once simplified, we find \( x = 11 \). This procedure of systematically performing operations to isolate the variable is the crux of solving linear equations.

Checking Solutions

After finding a solution to an equation, it's critical to verify it by substituting it back into the original equation. This step ensures that no mistakes were made during the calculations.In our example, we believe that \( x = 11 \) is the solution. Substituting 11 back into the original equation gives:\[ 11 + 14 = 25 \]Solving this confirms that both sides of the equation are equal, verifying that our solution is indeed correct.Checking solutions is a vital practice in solving equations as it helps catch and correct any errors made. It ensures that the found solution actually satisfies the original equation.

Equation Simplification

Simplifying an equation involves performing operations to make it easier to handle and solve, all while maintaining its equality. Let's consider how simplification works in the equation \( x + 14 = 25 \).

• Initially, the given equation has a straightforward structure but requires simplifying both sides once the subtraction is performed.
• Subtracting 14 from both sides as an operation is a form of simplification as it removes unnecessary terms to focus on the variable.
• This leads us to \( x = 11 \), a much simpler expression that clearly shows the value of \( x \).

The goal with simplification is to reduce complexity while maintaining accuracy, which facilitates easier understanding and solving of the equation at hand.