Problem 100 - Introductory Algebra for College Students

Question

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. When I substituted 5 for \(x\) in the equation $$4 x+6=6(x+1)-2 x$$ I obtained a true statement, so the equation's solution is \(5 .\)

Step-by-Step Solution

Answer

The short answer is that the statement 'does not make sense' because when substituting \(x = 5\) into the equation, the left hand side is not equal to the right hand side.'

Step 1: Substitute x

Replace \(x\) with \(5\) in the equation \(4x + 6 = 6(x + 1) - 2x\). The equation becomes \(4*5 + 6 = 6*(5 + 1) - 2*5\).

Step 2: Simplify the Equation

Solve and simplify both sides of the equation. This results in \(26 = 24\).

Step 3: Final Verdict

Since \(26\) is not equal to \(24\), the statement 'When I substituted 5 for \(x\) in the equation \(4x + 6 = 6(x + 1) - 2x\), I obtained a true statement, so the equation's solution is \(5\)' does not make sense.

Key Concepts

Substitution Method

The substitution method is a fundamental technique in algebra used to solve equations, particularly systems of equations. This approach involves replacing variables with specific values to determine if they satisfy the equation. In the context of the given problem, when you 'substitute 5 for x' in the algebraic expression, you are essentially checking whether 5 is a solution by seeing if the equation holds true after the replacement.

Here is how you perform the substitution step by step:

• Identify the variable in the equation (in this case, it's x).
• Replace every instance of this variable with the value you are testing (5 in this case).
• Perform arithmetic operations to see if both sides of the equation result in the same number after the substitution.

If, after the substitution, both sides of the equation are equal, the number is a solution to the equation. If not, the equation remains unsolved with respect to that number.

Equation Simplification

Equation simplification is a pivotal process in algebra that makes complex equations easier to solve. Simplification might include expanding parentheses, combining like terms, or reducing fractions to their simplest form. In our exercise, once the number 5 is substituted for x, the student is tasked with simplifying the resulting arithmetic equation.

To simplify effectively:

• Start by dealing with any parentheses by using the distributive property to eliminate them.
• Combine like terms on each side of the equation. These are terms that have the same variable raised to the same power.
• Perform any necessary arithmetic such as addition, subtraction, multiplication, or division to both sides of the equation to isolate the variable.

It's essential to apply these steps correctly to determine whether the equation is true or false and to avoid arriving at an incorrect conclusion.

Algebraic Reasoning

Algebraic reasoning is the process of justifying the solutions for algebraic equations based on logical deductions. This skill is crucial in detecting and correcting errors, understanding the relationships between quantities, and predicting the behavior of variables. When you analyze the process of substitution and simplification of the given equation, you apply algebraic reasoning to determine whether the statement provided is logical.

Effective algebraic reasoning involves:

• Understanding the properties of equality and operations.
• Knowing how and when to apply various algebraic rules and concepts.
• Checking the solution thoroughly to ensure it makes sense within the context of the problem.

In the example, algebraic reasoning is used in the final verdict to conclude that since after substitution and simplification the two sides of the equations are not equal, the original statement about the solution being 5 does not make sense.