Problem 28 - Algebra 1: Concepts and Skills

Question

Check to see if x = 5 is or is not a solution of the equation or the inequality. $$ 5 x \div 2=12.5 $$

Step-by-Step Solution

Answer

Yes, \(x = 5\) is a solution to the equation \(5 x \div 2=12.5\)

Step 1: Rearranging the equation

The equation given is \(5x \div 2=12.5\). To make it more clear, it can be rewritten as \(x = \frac{12.5 \times 2}{5}\).

Step 2: Simplifying the equation

Calculating the value on the right side of the equation, we get \(x = \frac{25}{5} = 5\).

Step 3: Verify the solution

Now, we substitute \(x = 5\) in the equation \(5x \div 2=12.5\), if both sides of the equation are balanced, then x = 5 is a solution. After substitution, we get \(5*5 \div 2=12.5\). Simplifying, we get 12.5 = 12.5, which is true.

Key Concepts

Understanding Algebra

Algebra is a branch of mathematics that uses symbols, typically letters, to represent numbers in equations or formulas. This method allows for the abstraction and manipulation of numbers to solve problems. In an algebraic equation, the goal is usually to find the value of the unknown. The unknown can be anything we want to find out.

For example, in the equation:

• \(5x \div 2 = 12.5\)

\(x\) stands for the unknown number we are trying to determine. Algebra simplifies complex problems by using these symbols to represent and systematically solve for unknown quantities. It's a fundamental skill used in everyday life, from calculating the best deals to understanding scientific data.

Exploring the Substitution Method

The substitution method is a powerful technique used in algebra to solve equations. It involves replacing a variable in an equation with a given value to simplify solving processes. This method is especially useful when checking if a certain value is a solution to a given equation or inequality.

Here's how it works, using our example:

• We are given the equation \(5x \div 2 = 12.5\).
• To check if \(x = 5\) is a solution, we substitute 5 into the equation.
• Rewriting the equation: \(5 \cdot 5 \div 2 = 12.5\). This is the substitution step.
• Simplify to verify: If both sides of the equation equal, then our solution is correct.

The substitution method is intuitive and provides a straightforward path to verifying potential solutions, as shown when \(x = 5\) solved the equation correctly.

The Importance of Verification in Equations

Verification is the last critical step in solving equations to ensure the solution is correct. Even if the mathematics seem right, checking your solution can prevent errors and confirm that the answer truly satisfies the original equation.

In our example:

• After solving \(x = 5\), we substitute back into the original equation: \(5x \div 2 = 12.5\).
• If the left side equals the right side after substitution, then our solution is verified.

Checking the work step-by-step ensures accuracy. Substituting solutions back can unveil mistakes or confirm that every part of the process is aligned. Verification strengthens understanding by linking the solution back to the original problem.