Problem 68 - Elementary Algebra

Question

For Problems 55-70, solve each equation for the indicated variable. (Objective 4) $$ \frac{x-3}{6}=\frac{y-4}{8} \text { for } x $$

Step-by-Step Solution

Answer

\( x = \frac{3y}{4} \)

Step 1: Understand the equation

We are given the equation \( \frac{x-3}{6} = \frac{y-4}{8} \) and we need to solve for \( x \). This means we have to manipulate the equation in such a way that \( x \) is isolated on one side.

Step 2: Eliminate the fractions

To eliminate the fractions, multiply both sides of the equation by 48, the least common multiple of 6 and 8: \[ 48 \times \frac{x-3}{6} = 48 \times \frac{y-4}{8} \]This simplifies to:\[ 8(x-3) = 6(y-4) \]

Step 3: Distribute the coefficients

Distribute the coefficients on both sides of the equation. This results in:\[ 8x - 24 = 6y - 24 \]

Step 4: Isolate the x-term

To bring \( x \) on one side, add 24 to both sides:\[ 8x = 6y \]

Step 5: Solve for x

Divide both sides of the equation by 8 to solve for \( x \):\[ x = \frac{6y}{8} \]Simplify the right side by dividing 6 and 8 by 2:\[ x = \frac{3y}{4} \]

Key Concepts

Solving for Variables

When solving an equation to find the value of a variable, the key goal is to isolate the variable on one side of the equation. In the case of the original exercise, we needed to manipulate the provided equation so that it expressed the solution in terms of the variable \( x \). The process typically involves using arithmetic operations to rearrange the equation. A systematic approach helps
make this easier and more efficient. Here's a simple way to think about it:

• Identify what is being asked: In this case, isolate \( x \).
• Use inverse operations: For additions, consider subtractions; for multiplications, think of divisions.
• Perform the same operation on both sides of the equation to maintain equality.
• Simplify step-by-step until the variable is alone on one side.

Understanding these steps not only solves the problem at hand but builds a foundation for tackling more complex algebraic equations.

Fraction Elimination

Fractions in an equation can complicate the process of isolating variables. Therefore, the first step often involves eliminating these fractions to simplify the equation. To do this, you need to find the Least Common Multiple (LCM) of the denominators. Multiply every term in the equation by this LCM:

• The LCM between 6 and 8 is 48.
• Multiplying both sides by 48 eliminates the fractions.
• Be cautious to multiply every part of the equation by this LCM.

The result will be a simplified equation without fractions, which becomes easier to solve for the variable. In the provided solution, multiplying by 48 transformed the equation into \( 8(x-3) = 6(y-4) \), which doesn't have fractions and allows for straightforward algebraic manipulation.

Equation Manipulation

After eliminating fractions, continue with basic algebraic operations to manipulate the equation. For equation manipulation, follow these strategies:

• Apply the distributive property where necessary: In \( 8(x-3) = 6(y-4) \), distribute to get \( 8x - 24 = 6y - 24 \).
• To solve for \( x \), bring all \( x \) terms to one side. In this step, add any constants to both sides: \( 8x = 6y \).
• Divide each term by the coefficient of \( x \) to isolate \( x \), resulting in \( x = \frac{6y}{8} \). Simplify to find \( x = \frac{3y}{4} \).

The ability to manipulate equations allows you to transition from a complex, dense equation into a simple form that directly provides the value for a variable. It's a core skill in algebra that can be applied to many different types of equations and problems.