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Chapter 2: Problem 48

Solve each equation. $$\frac{1}{3}(x-2)=\frac{2}{3} x-\frac{13}{3}$$

Short Answer

Expert verified
x = 11.

Step by step solution

01

Eliminate the fractions

Multiply both sides of the equation by 3 to eliminate the fractions. This gives: \(\frac{1}{3}(x-2) \times 3 = \frac{2}{3} x \times 3 - \frac{13}{3} \times 3\). Simplifying this yields: \(x - 2 = 2x - 13\).
02

Isolate the variable

Subtract \(2x\) from both sides to get all x terms on one side: \(x - 2 - 2x = 2x - 13 - 2x\). Simplify this to: \(-x - 2 = -13\).
03

Solve for x

Add 2 to both sides to isolate \(-x\): \(-x - 2 + 2 = -13 + 2\), which simplifies to: \(-x = -11\). Then, multiply both sides by -1 to solve for \(x\): \(x = 11\).

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Key Concepts

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

Eliminating Fractions
When solving equations, fractions can complicate the process. Eliminating fractions makes the equation easier to work with.

To eliminate fractions, multiply every term in the equation by the least common denominator (LCD). For example, in the equation \(\frac{1}{3}(x-2) = \frac{2}{3} x - \frac{13}{3}\), the denominator is 3.

Multiplying both sides by 3:
\[ \frac{1}{3}(x-2) \times 3 = \frac{2}{3} x \times 3 - \frac{13}{3} \times 3 \]
Simplifies to:
\[ x - 2 = 2x - 13 \]
Without fractions, it's easier to isolate the variable and solve the equation.
Isolating the Variable
Once fractions are eliminated, we need to isolate the variable. This means getting the variable on one side of the equation and non-variable terms on the other.

Starting from: \ x - 2 = 2x - 13 \
Subtract 2x from both sides to get rid of the x term on the right:
\[ x - 2 - 2x = 2x - 13 - 2x \]
This simplifies to:
\[ -x - 2 = -13 \]
Now the variable is isolated on one side, making it easier to solve for it.
Simplifying Equations
The final step is to simplify the equation to solve for the variable.

From the isolated variable equation: \ -x - 2 = -13 \
Add 2 to both sides to reduce the constant term:
\[ -x - 2 + 2 = -13 + 2 \]
This simplifies to:
\[ -x = -11 \]
Finally, multiply both sides by -1 to solve for x:
\[ x = 11 \]
Simplifying equations step-by-step helps to clearly solve for the unknown variable.

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Most popular questions from this chapter

Show a complete solution to each problem. Really odd integers. If the smaller of two consecutive odd integers is subtracted from twice the larger one, then the result is \(13 .\) Find the integers.

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