Problem 16 - Introductory Algebra for College Students

Question

Solve each equation using the addition property of equality. Be sure to check your proposed solutions. $$-13=x+11$$

Step-by-Step Solution

Answer

Based on the solution steps, the solution to the equation \(-13 = x + 11\) is \(x = -24\).

Step 1: Analyze the equation

Looking at the equation, \(-13 = x + 11\), you can see it's a simple linear equation where x is added to 11 where the result is -13.

Step 2: Application of addition property of equality

When solving for x, you want to isolate x on one side of the equation. In order to do this, you will need to subtract 11 from both sides of the equation: \(-13 - 11 = x + 11 - 11\). This simplifies to \(-24 = x\).

Step 3: Verifying the solution

You can confirm this solution by substituting x with -24 in the original equation and see if the equation holds true: \(-13 = -24 + 11\). Simplifying the right side: \(-13 = -13\). This shows that our solution is correct because both sides of the equation are equal.

Key Concepts

Linear Equations

Linear equations are equations of the first degree, meaning they involve only linear terms. These equations are called "linear" because, when graphed, they produce a straight line. A basic linear equation looks like this: \(ax + b = c\), where \(a\), \(b\), and \(c\) are constants.
To solve linear equations, one primary goal is to find the value of the variable that makes the equation true. In the given problem, we have the equation \(-13 = x + 11\), where our goal is to solve for \(x\). This specific equation includes:

• Constant Terms: These are numbers without any variables attached, such as \(-13\) and \(+11\).
• Variable Terms: In our equation, \(x\) is the variable term.

By focusing on isolating the variable, we aim to achieve an equation where \(x\) stands alone on one side.

Solving Equations

Solving equations involves manipulating them in such a way that you isolate the variable on one side of the equation. For linear equations, the addition property of equality is a fundamental tool.
The addition property of equality states that you can add or subtract the same value from both sides of an equation without changing its equality. For instance, in the equation \(-13 = x + 11\), we needed to remove the 11 on the same side as \(x\).

This gives us the rewritten expression:
\(-13 - 11 = x + 11 - 11\).
By simplifying, we find \(-24 = x\). Thus, the variable \(x\) is solution of the equation.
Key steps in solving such equations include:

• Identify the term to move: Look for constants or other terms that you need to move across the equality sign to help isolate the variable.
• Apply operations carefully: Use addition or subtraction to eliminate terms as necessary.
• Simplify: Combine like terms and simplify expressions to arrive at the final result.

Verification of Solutions

After solving an equation, it's crucial to ensure the solution is valid. This involves verifying that the value found for the variable satisfies the original equation. This step increases the confidence in the correctness of the solution.
For verification, take the calculated solution and substitute it back into the original equation. In our exercise, substituting \(x = -24\) into \(-13 = x + 11\) results in \(-13 = -24 + 11\).
Perform the arithmetic to confirm both sides match:

• The arithmetic on the right side simplifies to \(-13\), matching the left-hand side.
• This confirms that the calculated value of \(x\) is indeed correct.

Verification acts as a safeguard against errors, ensuring that the derived solution truly satisfies the equation. Always perform this step when solving equations to avoid mistakes. Ensuring accuracy is especially important in mathematical problem-solving.