Problem 105 - Introductory and Intermediate Algebra for College Students 4th

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The equation \(x+\frac{1}{3}=\frac{1}{2}\) is equivalent to \(x+2=3\).

Step-by-Step Solution

Answer

The statement 'The equation \(x + \frac{1}{3} = \frac{1}{2}\) is equivalent to \(x + 2 = 3\)' is false. For it to become a true statement, it must be changed to 'The equation \(x + \frac{1}{3} = \frac{1}{2}\) is equivalent to \(x + \frac{1}{3} = \frac{1}{2}\)'.

Step 1: Simplification

Begin by simplifying both equations. The first equation is \(x + \frac{1}{3} = \frac{1}{2}\). Since we want to isolate \(x\) in the equation, subtract \(\frac{1}{3}\) from both sides, which will result in \(x = \frac{1}{2} - \frac{1}{3}\). By finding a common denominator and then subtracting, it results in \(x = \frac{1}{6}\). The same simplification needs to be carried out with the second equation \(x + 2 = 3\). Subtract 2 from both sides to achieve \(x = 1\).

Step 2: Comparison

Upon comparing \(x\) in the two equations, it becomes clear that the value of \(x\) in first equation (\(x = \frac{1}{6}\)) is not equal to the value of \(x\) in the second equation (\(x = 1\)). So, these two equations are not equivalent.

Step 3: Proposal for Correction

Since the statement is false, we need to identify the change which will make it true. If we replace \(2\) with \(\frac{1}{3}\) in the second equation, we will get \(x + \frac{1}{3} = 3\). Then the simplification will result in \(x = \frac{8}{3}\), which doesn't equal \(x\) from the first equation. So, the necessary change would be to replace both \(2\) and \(3\) in the second equation to produce a new equivalent equation \(x + \frac{1}{3} = \frac{1}{2}\).

Key Concepts

Equation Simplification

When dealing with algebraic equations, one of the fundamental skills is knowing how to simplify equations effectively. Simplifying means making an equation easier to solve by rearranging it or reducing it in some way. In our exercise, we simplified the first equation, \(x + \frac{1}{3} = \frac{1}{2}\), by isolating \(x\) on one side of the equation. To do this, we subtracted \(\frac{1}{3}\) from both sides. This can be thought of as balancing a scale, where what you do to one side, you must do to the other.

• Start by determining the operation needed to isolate the variable. In this case, subtraction was necessary to remove the fraction from \(x\).
• Simplify any fractions by finding a common denominator to combine or subtract them. Here, \(\frac{1}{2} - \frac{1}{3}\) requires converting them to have the same denominator: \(\frac{3}{6} - \frac{2}{6}\), resulting in \(\frac{1}{6}\).
• Check your work by ensuring all operations have been applied correctly.

When you simplify equations, you're breaking them down into more manageable parts, often leading to the solution.

Equivalent Equations

Understanding when two equations are equivalent is critical in algebra. Equivalent equations produce the same value for the variable after simplification. In our case, we found that the given equations, \(x + \frac{1}{3} = \frac{1}{2}\) and \(x + 2 = 3\), are not equivalent because they do not simplify to the same value of \(x\).

• Equivalent equations will always have the same solution for \(x\). Here, the first equation simplifies to \(x = \frac{1}{6}\), while the second simplifies to \(x = 1\).
• To determine equivalence, simplify both equations fully and compare their solutions.
• Remember that different operations can affect equivalence. Adding, subtracting, or even fractions entails careful handling to maintain the integrity of the equations.

If two equations are not equivalent, like in this scenario, changes need to be made to align them or make clear why they're different.

Problem Solving Steps

Solving problems systematically helps solve equations easily and without errors. Let's breakdown the approach used in our exercise. Knowing how to navigate through each step with clarity is important.

• Step 1: Simplification - Simplify each equation independently to find the value of the variable \(x\). This involves arithmetic operations and possibly working with fractions.
• Step 2: Comparison - Compare the solutions of the simplified equations. Determine if the solved values for \(x\) are the same, indicating equivalence.
• Step 3: Proposal for Correction - If the equations are not equivalent but equivalence is needed, adjust the terms to make them so. This could involve altering constants or coefficients in the equation.
• Throughout each step, ensure all calculations are checked and logical deductions made. Mistakes often occur in detailed calculations.

Following these steps methodically can lead to clarity and accuracy in solving algebra problems, making each phase manageable and efficient.