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Chapter 8: Problem 341

In the following exercises, solve each equation. $$ \frac{1}{4} p+\frac{1}{3}=\frac{1}{2} $$

Short Answer

Expert verified
p = \( \frac{2}{3} \)

Step by step solution

01

Eliminate the fractions

To make it easier to solve, eliminate the fractions by finding the least common multiple (LCM) of the denominators (4, 3, and 2). The LCM is 12. Multiply every term in the equation by 12: \[ 12 \times \left( \frac{1}{4} p \right) + 12 \times \left( \frac{1}{3} \right) = 12 \times \left( \frac{1}{2} \right) \]This simplifies to: \[ 3p + 4 = 6 \]
02

Isolate the variable term

Subtract 4 from both sides of the equation to isolate the variable term: \[ 3p + 4 - 4 = 6 - 4 \]This simplifies to: \[ 3p = 2 \]
03

Solve for the variable

Divide both sides of the equation by 3 to solve for the variable p: \[ \frac{3p}{3} = \frac{2}{3} \]This simplifies to: \[ p = \frac{2}{3} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

prealgebra
Prealgebra is the foundation of all advanced math topics. It involves basic arithmetic operations like addition, subtraction, multiplication, and division. In prealgebra, we also deal with basic number properties, integers, fractions, decimals, and simple equations. This topic prepares students for algebra by introducing them to variables and simple equations.
least common multiple
The least common multiple (LCM) is the smallest number that is a multiple of two or more numbers. For example, the LCM of 4, 3, and 2 is 12. Finding the LCM is useful when working with fractions since it helps to eliminate the fractions from equations. Multiplying each term by the LCM ensures that all the denominators are eliminated. In our exercise, we multiplied each term by 12 (the LCM of 4, 3, and 2). This transformed the equation from fractions to whole numbers, making it easier to solve.
isolating variables
Isolating the variable is a crucial step in solving equations. It simply means rearranging the equation so that the variable is by itself on one side of the equation. After eliminating the fractions in our exercise, we had the equation 3p + 4 = 6. To isolate 'p', we subtracted 4 from both sides, resulting in 3p = 2. This step helps to simplify the equation further and prepares it for the final step of solving for the variable.
fractions in equations
Fractions can be tricky in equations, but there are ways to handle them effectively. One common method is to eliminate the fractions by finding the least common multiple of the denominators and multiplying the entire equation by it. This transforms the fractions into whole numbers. In our example equation \(\frac{1}{4} p + \frac{1}{3} = \frac{1}{2}\), we found the LCM of 4, 3, and 2 to be 12, and multiplied everything by 12 to eliminate the fractions. This led us to a much simpler equation without fractions, making it easier to solve.

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