Question

Solve the equation.
two sevenths plus one third x minus one fourth equals two thirds x
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Part 1
The solution is xequals
  
enter your response here.
​(Type an integer or a simplified​ fraction.)

Answer 1

To solve the equation

\[
\frac{2}{7} + \frac{1}{3}x - \frac{1}{4} = \frac{2}{3}x,
\]

first, we can rearrange it to combine like terms. Let's move all the terms involving \(x\) to one side and the constant terms to the other side:

\[
\frac{1}{3}x - \frac{2}{3}x = -\frac{2}{7} + \frac{1}{4}.
\]

This simplifies to:

\[
-\frac{1}{3}x = -\frac{2}{7} + \frac{1}{4}.
\]

We need to find a common denominator to combine the fractions on the right side. The least common multiple (LCM) of \(7\) and \(4\) is \(28\):

1. Convert \(-\frac{2}{7}\) to have a denominator of \(28\):
\[
-\frac{2}{7} = -\frac{2 \cdot 4}{7 \cdot 4} = -\frac{8}{28}.
\]

2. Convert \(\frac{1}{4}\) to have a denominator of \(28\):
\[
\frac{1}{4} = \frac{1 \cdot 7}{4 \cdot 7} = \frac{7}{28}.
\]

Now, combine the fractions:

\[
-\frac{8}{28} + \frac{7}{28} = -\frac{8 - 7}{28} = -\frac{1}{28}.
\]

So now we have:

\[
-\frac{1}{3}x = -\frac{1}{28}.
\]

To isolate \(x\), multiply both sides by \(-3\):

\[
x = -3 \left(-\frac{1}{28}\right) = \frac{3}{28}.
\]

Thus, the solution is:

\[
\boxed{\frac{3}{28}}.
\]