Problem 100 - Introductory and Intermediate Algebra for College Students 4th

Question

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. I'm solving a problem that requires me to determine if 5 is a solution of \(4 x+7\)

Step-by-Step Solution

Answer

The statement 'does not make sense' because the solution to the equation \(4x + 7\) when 'x' is '5' is '27', not '5'.

Step 1: Understand the Equation

In the equation \(4x + 7\), 'x' is the variable. To find out if '5' is a solution to the equation, replace 'x' with the value '5'.

Step 2: Substitute the value of 'x' in the equation

Now, substitute 'x' with '5' in the equation. The equation becomes: \(4*5 + 7\).

Step 3: Evaluate the Equation

Evaluate the equation \(4*5 + 7\). On calculation, the result is 27.

Step 4: Evaluate the Statement

The statement was whether '5' is a solution of the equation \(4x + 7\). After evaluating the equation, we find the solution as '27', not '5'. Therefore, the statement 'does not make sense'.

Key Concepts

Solution Verification

Verifying a solution in algebra is about checking if a specific value satisfies the equation. To determine if a certain number is a solution, you need to substitute it back into the original equation. After performing the substitution, if both sides of the equation are equal, the number is indeed a solution.

If they aren't equal, the number is not a solution. For example, when checking if 5 is a solution for the equation \(4x + 7\), substitute 5 for \(x\). You then calculate \(4 \times 5 + 7 \). If the result equals the original equation's right-hand side, then 5 is a solution. However, since \(4 \times 5 + 7 = 27\) and not the original equation's value, 5 is not a solution. Understanding and verifying solutions is crucial in ensuring accuracy in algebraic calculations.

Substitution Method

The substitution method involves replacing variables in an equation or expression with given values to solve or simplify. This method is essential when determining whether a number is a possible solution for an equation.

Here's how it works:

• Identify the variable(s) in your equation.
• Replace the variable(s) with the given numerical value.
• Perform the arithmetic operation to simplify.

In our example, \(x\) is replaced with 5 in the equation \(4x + 7\), transforming it to \(4 \times 5 + 7\). This process helps you "test" the number against the equation. Practicing substitution ensures you understand how numbers interact within algebraic equations.

Equation Evaluation

Equation evaluation is the process of calculating the numerical value of an equation after substituting the values into it. This process involves arithmetic operations that highlight whether a proposed solution meets the equation's requirements.

When you evaluate \(4 \times 5 + 7\), you perform multiplication first due to the order of operations (PEMDAS/BODMAS) and then the addition. Here, first calculate \(4 \times 5 = 20\) and then add 7 to get 27. This confirms whether your calculation was correctly done. The purpose of evaluation is to ensure the arithmetic correctness and logical consistency of potential solutions for algebraic equations.