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Chapter 2: Problem 328

Solve each equation with fraction coefficients. $$x+\frac{1}{2}=\frac{2}{3} x-\frac{1}{2}$$

Short Answer

Expert verified
The solution is \( x = -3 \).

Step by step solution

01

- Isolate the variable on one side

Start by getting all terms containing the variable on one side of the equation. Subtract \frac{2}{3} x\ from both sides.\[x + \frac{1}{2} - \frac{2}{3}x = \frac{2}{3}x - \frac{2}{3}x - \frac{1}{2}\]which simplifies to:\[x - \frac{2}{3}x + \frac{1}{2} = - \frac{1}{2}\]
02

- Combine like terms

Combine the \x terms on the left side. Note that \x is equivalent to \frac{3}{3}x\, so the equation becomes:\[\frac{3}{3}x - \frac{2}{3}x + \frac{1}{2} = -\frac{1}{2}\]Simplify the left side:\[\frac{1}{3}x + \frac{1}{2} = -\frac{1}{2}\]
03

- Isolate the variable term

Subtract \frac{1}{2}\ from both sides to isolate \frac{1}{3}x\:\[\frac{1}{3}x + \frac{1}{2} - \frac{1}{2} = -\frac{1}{2} - \frac{1}{2}\]Which simplifies to:\[\frac{1}{3}x = -1\]
04

- Solve for x

To isolate \field{x}\, multiply both sides of the equation by 3:\[\frac{1}{3}x \times 3 = -1 \times 3\]This simplifies to:\[x = -3\]

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Key Concepts

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

Variable Isolation
Isolating the variable is a crucial step in solving equations. This means getting all terms containing the variable on one side of the equation. In our exercise, we start with the equation: $$x + \frac{1}{2} = \frac{2}{3}x - \frac{1}{2}$$ To isolate the variable, subtract \frac{2}{3}x from both sides: $$x + \frac{1}{2} - \frac{2}{3}x = \frac{2}{3}x - \frac{2}{3}x - \frac{1}{2}$$ Combining these terms gets all the variables on one side and simplifies to: $$x - \frac{2}{3}x + \frac{1}{2} = -\frac{1}{2}$$
Combining Like Terms
Combining like terms helps to simplify equations so that we can solve them more easily. In our case, we need to combine the \field{x} terms. First, recognize that \field{x} is the same as \field{\frac{3}{3}x}: $$\frac{3}{3}x - \frac{2}{3}x + \frac{1}{2} = -\frac{1}{2}$$ Now, combine the x terms: $$\frac{1}{3}x + \frac{1}{2} = -\frac{1}{2}$$ This simplifies the equation and makes the variable term more manageable.
Solving for x
After simplifying, we need to solve for \field{x}. This means we need to isolate \field{x} completely. From the previous step, we have: $$\frac{1}{3}x + \frac{1}{2} = -\frac{1}{2}$$ Subtract \field{\frac{1}{2}} from both sides: $$\frac{1}{3}x + \frac{1}{2} - \frac{1}{2} = -\frac{1}{2} - \frac{1}{2}$$ This simplifies to: $$\frac{1}{3}x = -1$$
Multiplying Fractions
Multiplying fractions is often necessary when solving for a variable. To get \field{x} alone, we need to multiply both sides of our simplified equation by 3: $$\frac{1}{3}x \times 3 = -1 \times 3$$ Multiplying fractions involves multiplying the numerator (top number) and the denominator (bottom number) separately. Here, you multiply \frac{1}{3} by 3: $$\frac{1 \times 3}{3} = 1 \text{ and } -1 \times 3 = -3$$ So, \field{x} is finally: $$x = -3$$ Notice that multiplying both sides by the same number keeps the equation balance.

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