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Chapter 2: Problem 341

Solve each equation with fraction coefficients. $$\frac{1}{3}(x+5)=\frac{5}{6}$$

Short Answer

Expert verified
\( x = -\frac{5}{2} \)

Step by step solution

01

Eliminate the Fraction

Multiply both sides of the equation by the denominator of the fraction coefficient to clear the fractions. In this case, multiply both sides by 3. \(\frac{1}{3}(x+5) \times 3 = \frac{5}{6} \times 3\)This simplifies to: \(x + 5 = \frac{15}{6}\)
02

Simplify the Right Side

Simplify the fraction \( \frac{15}{6} \) on the right side of the equation. This simplifies to \( \frac{5}{2} \).So the equation now reads: \( x + 5 = \frac{5}{2} \)
03

Isolate the Variable

Subtract 5 from both sides of the equation to isolate \( x \). \( x + 5 - 5 = \frac{5}{2} - 5 \)Simplify the right side: \( x = \frac{5}{2} - \frac{10}{2} \)Which gives us: \( x = -\frac{5}{2} \)

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Key Concepts

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

Fraction Coefficients
When solving equations with fraction coefficients, the first thing you might notice is that it looks a bit more complicated than equations with whole numbers. A fraction coefficient is simply a number in fraction form that multiplies the variable in the equation. For example, in the equation \(\frac{1}{3}(x+5)=\frac{5}{6}\), \(\frac{1}{3}\) is the fraction coefficient.
Dealing with fraction coefficients can often make the equation less straightforward. However, the overall goal remains to isolate the variable. This means you will likely need to perform additional steps like multiplying or dividing by the denominator to clear the fractions.
Clearing Fractions
Clearing fractions means transforming the equation so that there are no fractions involved. This helps simplify the solving process. To clear fractions from an equation, you multiply all terms by the least common denominator (LCD) of all the fraction coefficients in the equation.
In our example equation: \(\frac{1}{3}(x+5)=\frac{5}{6}\), the denominator of the fraction coefficient \(\frac{1}{3}\) is 3. To clear the fraction, we multiply both sides by 3:
\(\frac{1}{3}(x+5) \times 3 = \frac{5}{6} \times 3\)
This gives us: \((x+5) = \frac{15}{6}\)
which simplifies to \(x + 5 = \frac{5}{2}\)
Now, the equation is much simpler to solve since it does not contain any fractions.
Isolating the Variable
The final main concept is isolating the variable. Once you have cleared the fractions, you need to isolate the variable to find its value. Let's look at our simplified equation \(x + 5 = \frac{5}{2}\)
To isolate x, subtract 5 from both sides:
\(x + 5 - 5 = \frac{5}{2} - 5\)
Simplify the equation step-by-step. First, which:
\(x = \frac{5}{2} - \frac{10}{2}\)
This simplifies to:
\(x = -\frac{5}{2}\)
By isolating the variable, you've now successfully solved for x.

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