Let's solve the equation step-by-step:
Given:
[tex]\[
\frac{3}{7} + \frac{1}{5}x - \frac{1}{4} = \frac{4}{5}x
\][/tex]
1. Combine the like terms on the left-hand side:
[tex]\[
\frac{3}{7} - \frac{1}{4} + \frac{1}{5}x = \frac{4}{5}x
\][/tex]
2. Find a common denominator for the fractions [tex]\(\frac{3}{7}\)[/tex] and [tex]\(\frac{1}{4}\)[/tex]:
- The least common multiple (LCM) of 7 and 4 is 28.
[tex]\[
\frac{3}{7} = \frac{3 \times 4}{7 \times 4} = \frac{12}{28}
\][/tex]
[tex]\[
\frac{1}{4} = \frac{1 \times 7}{4 \times 7} = \frac{7}{28}
\][/tex]
3. Subtract the fractions:
[tex]\[
\frac{12}{28} - \frac{7}{28} = \frac{5}{28}
\][/tex]
Now substitute back into the equation:
[tex]\[
\frac{5}{28} + \frac{1}{5}x = \frac{4}{5}x
\][/tex]
4. Move all the [tex]\(x\)[/tex]-terms to one side by subtracting [tex]\(\frac{1}{5}x\)[/tex] from both sides:
[tex]\[
\frac{5}{28} = \frac{4}{5}x - \frac{1}{5}x
\][/tex]
[tex]\[
\frac{5}{28} = \frac{3}{5}x
\][/tex]
5. Solve for [tex]\(x\)[/tex]:
First, multiply both sides by 5 to clear the fraction on the right-hand side:
[tex]\[
5 \cdot \frac{5}{28} = 3x
\][/tex]
[tex]\[
\frac{25}{28} = 3x
\][/tex]
6. Divide both sides by 3 to isolate [tex]\(x\)[/tex]:
[tex]\[
x = \frac{25}{28 \cdot 3}
\][/tex]
[tex]\[
x = \frac{25}{84}
\][/tex]
Therefore, the solution is:
[tex]\[
x = \frac{25}{84}
\][/tex]
This simplifies to approximately:
[tex]\[
x \approx 0.297619047619048
\][/tex]
