Problem 4 - Elementary Algebra

Question

For Problems \(1-72\), use the properties of equality to help solve each equation. $$ x+13=2 $$

Step-by-Step Solution

Answer

The solution is \( x = -11 \).

Step 1: Identify the Equation

The equation we need to solve is given as \( x + 13 = 2 \). The goal is to determine the value of \( x \) that satisfies this equation.

Step 2: Isolate the Variable

To solve for \( x \), we need to isolate \( x \) on one side of the equation. We do this by subtracting 13 from both sides of the equation: \( x + 13 - 13 = 2 - 13 \).

Step 3: Simplify Both Sides

Simplify both sides of the equation: \( x = 2 - 13 \), which results in \( x = -11 \).

Step 4: Verify the Solution

Check your solution by substituting \( x = -11 \) back into the original equation to ensure it holds true: \(-11 + 13 = 2\), which is true.

Key Concepts

Solving Equations

When solving equations, we want to find the value of the variable that makes the equation true. An equation is like a balance scale. The left side must equal the right side for the balance to hold. In the example of the equation \( x + 13 = 2 \), our job is to determine what \( x \) is, so both sides give the same result. Here's how we go about solving it:

• Identify the equation and understand the components involved.
• Recognize what type of mathematical operations are needed to solve it.
• Apply mathematical operations that maintain the balance of the equation.

In this exercise, we deal with a linear equation. Linear equations are usually easier to solve because they only involve one variable raised to the power of one. They often follow a straightforward set of steps using the properties of equality to ensure that we establish equality on both sides after every step. This foundational understanding will guide you in tackling equations with confidence.

Isolating Variables

Isolating variables is crucial when working with equations because it allows you to simplify the problem and make it more manageable. The essence is to get the variable alone on one side of the equation. In the exercise \( x + 13 = 2 \), we had to isolate \( x \). Here’s the process:

• Identify what is added or subtracted with the variable.
• Use opposite operations to both sides to remove numbers or terms keeping the balance intact.

For example, since \( 13 \) is added to \( x \), we performed subtraction of \( 13 \) from both sides. Thus, the equation \( x + 13 = 2 \) became \( x = 2 - 13 \), ultimately leading to \( x = -11 \). The idea is to change the equation step by step while keeping it equivalent, by moving unwanted numbers or terms away from the variable. This isolating action serves a purpose: to simplify the equation until the unknown variable stands alone, making it easy to identify its exact value.

Algebraic Verification

Algebraic verification is a key final step when solving equations. It confirms that the solution you've derived satisfies the original equation. To verify, take the value you've found for the variable and substitute it back into the original equation to see if it holds true. This process prevents errors and ensures accuracy.Here's how verification works for \( x + 13 = 2 \):

• Substitute \( x = -11 \) back into the left-hand side of the original equation.
• Perform the arithmetic: \(-11 + 13\).
• The result should match the other side of the equation, which is \( 2 \).

In our case, \(-11 + 13\) does result in \( 2 \), thus confirming that \( x = -11 \) is indeed the correct solution. Without this verification step, we might miss potential errors. Hence, always recheck that the solution is valid by substituting it back into the equation. This practice reinforces understanding and boosts confidence in solving equations accurately.