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Magazine Chapter 9: Problem 20
If the exercise is an equation, solve it and check. Otherwise, perform the indicated operations and simplify. $$\frac{x}{2}+\frac{x}{3}=\frac{x}{4}$$
Short Answer
Expert verified
x = 0
Step by step solution
01
Find a common denominator
The given equation is \(\frac{x}{2} + \frac{x}{3} = \frac{x}{4}\). First, find the least common multiple (LCM) of 2, 3, and 4, which is 12. Rewrite each fraction with a denominator of 12.
02
Rewrite each fraction
Rewrite the fractions using the common denominator of 12: \(\frac{x}{2} = \frac{6x}{12}\), \(\frac{x}{3} = \frac{4x}{12}\), and \(\frac{x}{4} = \frac{3x}{12}\). The equation becomes \(\frac{6x}{12} + \frac{4x}{12} = \frac{3x}{12}\).
03
Combine the fractions
Combine the fractions on the left side of the equation: \(\frac{6x}{12} + \frac{4x}{12} = \frac{10x}{12}\). The equation is now \(\frac{10x}{12} = \frac{3x}{12}\).
04
Simplify the equation
Since the denominators are the same, you can set the numerators equal to each other: \(10x = 3x\).
05
Solve for x
Subtract \(3x\) from both sides of the equation: \(10x - 3x = 0\). This simplifies to \(7x = 0\).
06
Solve for x (continued)
Divide both sides by 7 to solve for x: \(x = 0\).
07
Check the solution
Substitute \(x = 0\) back into the original equation to verify the solution: \(\frac{0}{2} + \frac{0}{3} = \frac{0}{4}\), which simplifies to \(0 = 0\). The solution is correct.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
common denominator
Understanding how to find a common denominator is essential when working with fractions in equations. A common denominator allows you to combine fractions by converting them to have the same denominator. This process is necessary to simplify equations involving fractions.
To find a common denominator:
To find a common denominator:
- Identify the denominators of all fractions involved. For example, in the equation \(\frac{x}{2} + \frac{x}{3} = \frac{x}{4}\), the denominators are 2, 3, and 4.
- Calculate the least common multiple (LCM) of these numbers. The LCM of 2, 3, and 4 is 12.
- Rewrite each fraction so they all have the common denominator. This ensures all parts of the equation are comparable, which simplifies further calculations.
combining fractions
After finding a common denominator, the next step is combining fractions. When fractions have the same denominator, you can add or subtract them easily. This is a crucial skill for simplifying equations.
From our example equation:
\(\frac{10x}{12} = \frac{3x}{12}\)
From our example equation:
- First, rewrite each fraction using the common denominator of 12:
\(\frac{x}{2} = \frac{6x}{12}\), \(\frac{x}{3} = \frac{4x}{12}\), \(\frac{x}{4} = \frac{3x}{12}\). - Combine the fractions on the same side of the equation:
\(\frac{6x}{12} + \frac{4x}{12} = \frac{10x}{12}\).
\(\frac{10x}{12} = \frac{3x}{12}\)
least common multiple
Least Common Multiple (LCM) is a key concept when working with fractions. The LCM of two or more numbers is the smallest number that is a multiple of all the numbers. Finding the LCM is important to determine a common denominator.
To find the LCM:
To find the LCM:
- List the multiples of each number.
- Find the smallest multiple that all original numbers share.
- Multiples of 2: 2, 4, 6, 8, 10, 12, ...
- Multiples of 3: 3, 6, 9, 12, ...
- Multiples of 4: 4, 8, 12, ...
solving for variables
Finally, solving for variables involves isolating the variable on one side of the equation. This step-by-step approach unravels the value of the variable in question.
Applying these methods to our equation:
Applying these methods to our equation:
- After combining the fractions, we have \(\frac{10x}{12} = \frac{3x}{12}\).
- We then remove the denominator by focusing on the numerators: \(10x = 3x\).
- Simplify by moving terms involving \(x\) to one side:
\(10x - 3x = 0\), leading to \(7x = 0\). - Divide both sides by 7 to find the value of \(x\):
\(x = 0\).
