Problem 3 - Elementary Algebra

Question

For Problems \(1-72\), use the properties of equality to help solve each equation. $$ x+11=5 $$

Step-by-Step Solution

Answer

The solution is \( x = -6 \).

Step 1: Understanding the Equation

The given equation is \( x + 11 = 5 \). Our goal is to isolate \( x \) on one side of the equation to find its value.

Step 2: Subtract 11 from Both Sides

To isolate \( x \), subtract 11 from both sides of the equation:\[x + 11 - 11 = 5 - 11\]

Step 3: Simplify Both Sides

Simplify the left side where \( 11 \) is subtracted, leaving \( x \). On the right side, calculate \( 5 - 11 \):\[x = -6\]

Step 4: Verify the Solution

Substitute \( x = -6 \) back into the original equation to ensure it satisfies the equation:\[-6 + 11 = 5\]Simplifying, we get \( 5 = 5 \), which is true.

Key Concepts

Properties of Equality

Solving linear equations primarily revolves around manipulating the equation while preserving equality. This is where the properties of equality become essential tools. These properties allow us to transform an equation without altering its truthfulness.

Here are several core properties:

• Addition and Subtraction Property: If you add or subtract the same number from both sides of an equation, the equality remains intact. For example, in the equation \(x + 11 = 5\), subtracting 11 from both sides gives the modified equation \(x = -6\). This operation keeps the equation balanced.
• Multiplication Property: Multiplying both sides by the same non-zero number keeps the equality valid. It's useful in equations where the coefficient of the variable needs to be simplified.
• Division Property: Similarly, dividing both sides by the same non-zero number maintains equality, ideal when isolating a variable with a coefficient.

Understanding these properties helps you confidently manipulate equations, ensuring any changes or operations preserve the equation's balance. As demonstrated in the solution, using subtraction carefully adjusted the equation for solving.

Isolation of Variables

Isolating a variable means rearranging the equation to have the desired variable alone on one side. This is akin to unwrapping a present to see what's inside—you want to reveal the mystery value of the variable.

The simplest linear equations, like \(x + 11 = 5\), require straightforward maneuvers:

• To isolate \(x\), notice what's being added to it—11 in this case. Performing the opposite operation (subtraction) on both sides effectively "removes" the 11.
• Repeat the operation for clarity: \(x + 11 - 11 = 5 - 11\) results in \(x = -6\).

Once you've isolated the variable, you've unveiled its value. Isolation paves the way to uncovering the specific number each variable represents. It's a fundamental part of solving any equation and, when done correctly, provides a clear solution insight.

Verification of Solutions

After solving for a variable, the pivotal step is to verify that your solution is correct. Verification provides assurance that the manipulation of properties and operations was executed flawlessly.

Here's how you verify:

• Take your solution, such as \(x = -6\), and substitute it back into the original equation \(x + 11 = 5\).
• Replace \(x\) with \(-6\), resulting in \(-6 + 11\).
• Calculate: \(-6 + 11 = 5\), which holds true as consistency is maintained.

Verification is like the final stamp of approval, confirming that each step taken was correct and that the resulting value satisfies the initial equation. It solidifies your understanding and ensures the solution stands on firm mathematical ground. By integrating verification, you develop a practiced method for authenticating your solutions, boosting confidence in your problem-solving capabilities.