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Chapter 2: Problem 36

\(-\frac{1}{4}=x-\frac{2}{3}\)

Short Answer

Expert verified
x = \(\frac{5}{12}\)

Step by step solution

01

- Isolate the variable

Add \(\frac{2}{3}\) to both sides of the equation to isolate \(x\). This step is needed to move \(\frac{2}{3}\) to the left side, so the equation becomes: \(-\frac{1}{4} + \frac{2}{3} = x\).
02

- Find a common denominator

To add \(-\frac{1}{4}\) and \(\frac{2}{3}\), find a common denominator. The least common denominator of 4 and 3 is 12. Convert both fractions to have this common denominator: \(-\frac{1}{4} = -\frac{3}{12}\) and \(\frac{2}{3} = \frac{8}{12}\).
03

- Add the fractions

With the common denominator, add the fractions: \(-\frac{3}{12} + \frac{8}{12} = \frac{5}{12}\). Therefore, the equation simplifies to: \(x = \frac{5}{12}\).

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Key Concepts

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

isolating the variable
In a linear equation, you often need to isolate the variable to find its value. This means you get the variable by itself on one side of the equation. For example, in \(-\frac{1}{4}=x-\frac{2}{3}\), we want to move everything else away from \(x\).
To do this:
  • Add or subtract terms from both sides of the equation. This keeps the equation balanced.
  • In our example, we add \(\frac{2}{3}\) to both sides to remove it from the right side:
\(-\frac{1}{4} + \frac{2}{3} = x\)
This gives us an equation where \(x\) is isolated on one side, making it easier to solve.
common denominator
When adding or subtracting fractions, it's essential to have a common denominator. This is the same bottom number in both fractions. Finding a common denominator allows you to combine the fractions easily.
In our problem, \(-\frac{1}{4} + \frac{2}{3}\), the denominators are 4 and 3. The Least Common Denominator (LCD) of 4 and 3 is 12. So, we convert both fractions to have this common denominator:
  • \(-\frac{1}{4}\) becomes \-\frac{3}{12}\
  • \(\frac{2}{3}\) becomes \frac{8}{12}\
This step prepares the fractions so they can easily be added together.
adding fractions
Once the fractions have the same denominator, adding them is straightforward. You add the numerators and keep the denominator the same. In our example, we have:
  • \(-\frac{3}{12} + \frac{8}{12}\)
Add the numerators: -3 + 8 = 5, so:
\(\frac{5}{12}\)
Therefore, the original equation \(-\frac{1}{4}=x-\frac{2}{3}\) simplifies to \x = \frac{5}{12}\. The fractions are now added correctly, giving us the value of the isolated variable, \(x\).

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Most popular questions from this chapter

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