Problem 15 - Algebra 1

Question

Check whether the given number is a solution of the equation. $$\frac{x}{4}-7=13 ; 24$$

Step-by-Step Solution

Answer

No, 24 is not a solution to the given equation.

Step 1: Understand the Problem

We are given an equation \(\frac{x}{4}-7=13\) and a supposed solution for x, which is 24. The goal is to confirm whether 24 is indeed a solution to this equation or not.

Step 2: Substitute the Number

Let's plug x=24 back into the equation and try to simplify it. This gives us \(\frac{24}{4}-7\).

Step 3: Simplify the Result

Simplify the expression on the left. The result is 6-7 = -1.

Step 4: Compare the Result

Now compare this result with the right hand side of the equation. Clearly, -1 does not equal to 13. Therefore, x=24 is not a solution to the equation.

Key Concepts

Substitution Method

The substitution method is a technique often used to check whether a given value is a solution to an equation. It involves replacing the variable in the equation with the proposed solution and simplifying to see if both sides of the equation are equal. This is a systematic way to solve or verify solutions in algebraic problems.
In this exercise, we start with the equation \(\frac{x}{4}-7=13\) and a proposed solution \(x=24\). Using substitution:

• Replace \(x\) with \(24\) in the equation, leading to \(\frac{24}{4}-7\).
• Next, perform the division and subtraction operations as guided by arithmetic rules.
• Simplify to find the result \(6-7\), which results in \(-1\).

Thus, this method quickly tells us whether our guess or given number satisfies the equation. Substitution keeps our calculations clear and straightforward, reflecting logical consistency in evaluating whether a number is indeed a solution.

Algebraic Verification

Algebraic verification is the process of confirming that a solution is correct by using algebraic operations. It's a critical step after substitution, ensuring that the mathematical calculations are precise and logically sound.

In our case, after substituting \(x = 24\) into the equation \(\frac{x}{4}-7=13\), we need to verify the calculations:

• The division \(\frac{24}{4}\) results in \(6\).
• Perform the subtraction: \(6 - 7 = -1\).
• Compare this result \(-1\) with the right-hand side of the equation, which is \(13\).

The values do not match. Hence, algebraic verification confirms that \(x=24\) doesn't solve the equation. This highlights the importance of checking calculations as it prevents errors in determining solutions.

Equation Testing

Equation testing is the trial process through which we ascertain whether a specific value satisfies a given equation. This exercise demonstrates the broader testing process by applying logical steps to evaluate potential solutions.
When testing an equation:

• Start by substituting the test value, in this case, \(x = 24\), into the equation \(\frac{x}{4} - 7 = 13\).
• Follow up by calculating each term to reach a simplification following algebraic principles, particularly focusing on operations like division and subtraction.
• Determine whether both sides of the equation match.

Here, testing shows the simplification \(-1\) is not equal to \(13\), meaning \(x=24\) is not a solution. Equation testing is reliable for confirming or refuting potential solutions by ensuring each step is correctly aligned with the mathematical logic required in solving equations.