Problem 35 - Algebra for College Students

Question

Solve each equation for \(x\). \(\frac{1}{3} x+a=\frac{1}{2} b\)

Step-by-Step Solution

Answer

The solution is \( x = \frac{3}{2} b - 3a \).

Step 1: Isolate the term containing x

Start with the equation \( \frac{1}{3} x + a = \frac{1}{2} b \). We need to isolate the term involving \( x \) on one side. Let's subtract \( a \) from both sides to achieve this: \( \frac{1}{3} x = \frac{1}{2} b - a \).

Step 2: Solve for x

Now that \( \frac{1}{3} x = \frac{1}{2} b - a \), we need to solve for \( x \) by removing the fraction. Multiply both sides by \( 3 \) to clear the fraction: \( x = 3 \left( \frac{1}{2} b - a \right) \). Distribute the \( 3 \) to both terms: \( x = \frac{3}{2} b - 3a \).

Key Concepts

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying equations to make them easier to solve. It is a fundamental skill in algebra that requires understanding several operations such as addition, subtraction, multiplication, and division.
One important aspect of algebraic manipulation is the ability to recognize like terms and perform operations to simplify an equation. For example, if an equation has multiple terms with the variable you're solving for, combining like terms is a helpful step. In the equation \( \frac{1}{3} x + a = \frac{1}{2} b \), the aim is to get all terms involving \( x \) on one side. This is achieved by subtracting \( a \) from both sides, resulting in \( \frac{1}{3} x = \frac{1}{2} b - a \).

• Identify what you need to isolate (here, \( x \)).
• Perform operations on both sides of the equation to maintain equality.

Understanding how to manipulate equations is key. It allows you to systematically simplify complex problems into simpler, more manageable steps.

Solving for a Variable

Solving for a variable involves isolating the variable in an equation. It means making the variable the subject of the formula, with all the other terms on the opposite side.
In the equation \( \frac{1}{3} x = \frac{1}{2} b - a \), the next step is to solve for \( x \). To do this, we need to eliminate the fraction by multiplying both sides by the denominator, which is 3 in this case. This results in \[ x = 3 \left( \frac{1}{2} b - a \right) \].
Understanding the balancing act of equations is crucial. You manipulate the equation to solve for \( x \):

• Use inverse operations to isolate the variable (e.g., multiply to eliminate division).
• Ensure operations are applied equally to both sides.

Finally, distribute the multiplier 3 across the terms to get \( x = \frac{3}{2} b - 3a \). This simplifies the equation and completes the process of solving for the variable.

Fractions in Equations

Handling fractions in equations is a skill needing clarity and confidence. Fractions can make equations look more complex, but with the correct approach, they are manageable.
In the original equation, \( \frac{1}{3} x + a = \frac{1}{2} b \), fractions represent parts of the whole. Dealing with them typically involves "clearing" the fractions by multiplying through by the denominator.

• Multiplying both sides by the least common denominator helps to eliminate fractions and simplify calculations.
• For example, multiply by 3 to clear the fraction in \( \frac{1}{3} x \) yielding \( x = 3 \left( \frac{1}{2} b - a \right) \).

Fractions are often a hurdle, but once cleared, they make equations easier to handle. Focus on practicing the elimination and manipulation, considering each fraction as a step toward isolating your variable.