Problem 21 - Elementary and Intermediate Algebra

Question

Solve each equation and check the result. If an equation has no solution, so indicate. $$ \frac{b}{4}+\frac{1}{2}=\frac{b}{3}-\frac{1}{4} $$

Step-by-Step Solution

Answer

The solution is \( b = 9 \). It is verified by substitution.

Step 1: Clear the Fractions

Multiply every term in the equation by the least common multiple of 4, 2, 3, and 4, which is 12, to clear the fractions: \[ 12 \left( \frac{b}{4} \right) + 12 \left( \frac{1}{2} \right) = 12 \left( \frac{b}{3} \right) - 12 \left( \frac{1}{4} \right) \]

Step 2: Simplify the Equation

Simplify each term: \[ 3b + 6 = 4b - 3 \]

Step 3: Move Terms to One Side

Subtract \( 3b \) from both sides to move the terms involving \( b \) to one side of the equation: \[ 6 = b - 3 \]

Step 4: Solve for \( b \)

Add 3 to both sides to solve for \( b \): \[ b = 9 \]

Step 5: Check the Solution

Substitute \( b = 9 \) back into the original equation to verify that it holds true:\[ \frac{9}{4} + \frac{1}{2} = \frac{9}{3} - \frac{1}{4} \] Calculate each side: - Left side: \( \frac{9}{4} + \frac{1}{2} = \frac{9}{4} + \frac{2}{4} = \frac{11}{4} \) - Right side: \( \frac{9}{3} - \frac{1}{4} = 3 - \frac{1}{4} = \frac{12}{4} - \frac{1}{4} = \frac{11}{4} \) Since both sides are equal, the solution \( b = 9 \) is verified.

Key Concepts

Fractions in Algebra

Fractions can often make algebraic equations seem more complex than they are. Understanding how to handle fractions is key to solving these equations. In algebra, fractions involve a numerator and a denominator, just like in arithmetic. But the difference is that variables can appear in either part.

When you encounter an equation with fractions, it's important to remember the rule of the least common multiple (LCM). The LCM is essential for clearing fractions, as it helps in converting all fractions to have the same denominator. This allows for easy addition, subtraction, or even multiplication and division.

• Pick the LCM of all denominators in the equation.
• Multiply every term in the equation by this LCM to eliminate the fractions.
• This step simplifies the equation and makes the rest of the solving process much easier.

By doing this, you effectively "clear" the fractions from the equation, which simplifies the solving process and lets you move forward with familiar algebraic operations.

Simplifying Equations

Once you have cleared an equation of fractions, simplifying it becomes much more straightforward. The goal of simplifying is to ensure the equation is in a form where you can easily solve for the unknown variable. This involves combining like terms and moving terms from one side of the equation to the other.

To simplify an equation, follow these steps:

• Begin by applying standard algebraic rules to combine like terms.
• Move all terms containing the variable to one side of the equation. This often involves adding or subtracting terms from both sides to "isolate" the variable.
• Lastly, perform any necessary operations to solve for the variable.

In our specific case, after clearing the fractions, we derived an equation without fractions: \( 3b + 6 = 4b - 3 \). Simplifying this required moving the \(3b\) term and isolating \(b\) on one side, which led us to the solution \( b = 9 \).

Checking Solutions

The final step in solving an algebra equation is checking your solution, which confirms the accuracy of your answer. By substituting the solution back into the original equation, you can verify whether both sides of the equation remain equal.

Here's the process:

• Substitute the found solution back into the original equation.
• Calculate both sides of the equation separately to ensure they equal each other.
• If they match, your solution is correct. If not, re-evaluate your steps for possible errors.

In our exercise, substituting \(b = 9\) into the original fractions showed that both left and right sides equaled \( \frac{11}{4} \). This balance confirmed that our solution was correct, ensuring no steps were missed in solving or simplifying the equation.