How do you write the least common multiple that clears an equation of fractions?
To write the least common multiple (LCM) that clears an equation of fractions, you need to find the least common denominator (LCD) of the fractions involved in the equation.
Here's how you can do this:
Step 1: Identify the denominators in the equation.
Look for all the denominators of the fractions in the equation. For example, if you have the equation:
\(\frac{2}{3}x + \frac{1}{4} = \frac{5}{6}\)
The denominators are 3, 4, and 6.
Step 2: Find the least common multiple (LCM) of the denominators.
To find the LCM, you need to list the multiples of each denominator and then find the least common multiple among them. Let's find the LCM for the denominators 3, 4, and 6.
Multiples of 3: 3, 6, 9, 12, 15, 18, ...
Multiples of 4: 4, 8, 12, 16, 20, 24, ...
Multiples of 6: 6, 12, 18, 24, 30, ...
The LCM is the smallest number that appears in all three lists. In this case, the LCM of 3, 4, and 6 is 12.
Step 3: Rewrite the equation using the LCM.
Now that you have the LCM, you need to rewrite the equation using the LCM as the common denominator. For each fraction in the equation, multiply both the numerator and the denominator by a necessary factor to obtain a denominator of 12.
\(\frac{2}{3}x + \frac{1}{4} = \frac{5}{6}\)
Multiply the first fraction by 4/4 and the second fraction by 3/3 to clear their denominators:
\(\frac{2}{3} \times \frac{4}{4}x + \frac{1}{4} \times \frac{3}{3} = \frac{5}{6} \times \frac{2}{2}\)
This results in:
\(\frac{8}{12}x + \frac{3}{12} = \frac{10}{12}\)
Now, the equation is clear of fractions.
Step 4: Solve the equation.
From here, you can solve the equation regularly by combining like terms, isolating the variable, and solving for it:
\(\frac{8}{12}x + \frac{3}{12} = \frac{10}{12}\)
\(\frac{8}{12}x = \frac{10}{12} - \frac{3}{12}\)
\(\frac{8}{12}x = \frac{7}{12}\)
\(x = \frac{7}{12} \div \frac{8}{12}\)
\(x = \frac{7}{12} \times \frac{12}{8}\)
\(x = \frac{7}{8}\)
So, the solution to the equation is \(x = \frac{7}{8}\), after rewriting the equation with the least common multiple that cleared the fractions.