Problem 9 - Elementary Algebra

Question

Solve each of the equations. $$\frac{x+1}{6}=\frac{x+2}{4}$$

Step-by-Step Solution

Answer

The solution is \(x = -4\).

Step 1: Understand the Problem

We have the equation \( \frac{x+1}{6} = \frac{x+2}{4} \) with two fractions set equal to each other, which implies a proportion that can be solved by cross-multiplying.

Step 2: Cross-Multiply

Cross-multiply the fractions to eliminate the denominators, resulting in the equation: \((x+1) \cdot 4 = (x+2) \cdot 6\).

Step 3: Distribute and Simplify

Distribute the numbers on both sides: \(4x + 4 = 6x + 12\).

Step 4: Move Terms Involving x

Subtract \(4x\) from both sides to get: \(4 = 2x + 12\).

Step 5: Isolate the Variable

Subtract 12 from both sides: \(4 - 12 = 2x\), simplifying to: \(-8 = 2x\).

Step 6: Solve for x

Divide both sides by 2 to solve for \(x\): \(x = -4\).

Step 7: Verify the Solution

Substitute \(x = -4\) back into the original equation to verify: \(\frac{-4+1}{6} = \frac{-4+2}{4}\) simplifies to \(\frac{-3}{6} = \frac{-2}{4}\), and both sides equal \(-\frac{1}{2}\), confirming the solution is correct.

Key Concepts

Proportion

A proportion indicates that two ratios are equal to each other. Think of it as a statement where two fractions are set to be the same, like our given equation \(\frac{x+1}{6} = \frac{x+2}{4}\). Both sides imply that when one numerator is divided by its corresponding denominator, the result should be the same as when the other numerator is divided by its corresponding denominator.

This can occur in various real-life scenarios, such as scaling a recipe, analyzing unit rates, or dealing with similar triangles in geometry.

In mathematics, understanding proportions helps you simplify complex comparisons and make predictions. Recognizing the presence of a proportion allows you to use specific methods like cross-multiplying to solve equations efficiently. This is a powerful tool because it converts a potentially complicated equation into a simpler one that's easier to manage.

Cross-multiplying

Whenever you deal with a proportion, like with our equation \(\frac{x+1}{6} = \frac{x+2}{4}\), cross-multiplying is a handy technique to clear the fractions. Essentially, you multiply across the diagonal.

• Multiply the numerator of one fraction by the denominator of the other fraction.
• Perform the same operation across the remaining diagonal.

This results in an equation without fractions. For the exercise, this step changes it to \((x+1) \cdot 4 = (x+2) \cdot 6\).

This process not only simplifies the terms but also keeps the balance of the equation intact. It eliminates the denominators, allowing you to focus solely on the polynomials. This method is especially useful in algebra because it allows listeners to tackle otherwise intimidating equations part by part, maintaining clarity and conciseness throughout the solving process.

Solving Equations

Solving equations, such as the one we derived from cross-multiplying, involves finding the value of the variable that makes the equation true. Let's look at our simplified equation: \(4x + 4 = 6x + 12\). Here’s a step-by-step approach:

• Distributing: Both sides are expanded if needed, which already was done in our case.
• Simplifying: Begin by solving for \(x\): remove or combine like terms and constants from both sides. For our equation:
  - Subtract \(4x\) from both sides to keep terms involving \(x\) on one side: \(4 = 2x + 12\).
• Isolating the Variable: Remove constants next to \(x\) by reversing any addition or subtraction: subtract 12 from both sides resulting in \(-8 = 2x\).
• Finalizing the Solution: Divide by the coefficient of \(x\) to isolate \(x\): \(x = -4\).

In problem-solving, these steps are building blocks to find unknowns by isolating variables. Once you find a potential solution, always check back with the original equation to ensure accuracy. In our case, substituting \(x = -4\) back confirms that the equation holds true on both sides.