To solve the equation [tex]\(\frac{3}{5}(x-4)+\frac{6}{7}=\frac{1}{7}(x+2)\)[/tex], we can follow these steps:
1. Distribute and simplify both sides:
[tex]\[
\frac{3}{5}(x - 4) + \frac{6}{7} = \frac{1}{7}(x + 2)
\][/tex]
Distribute [tex]\(\frac{3}{5}\)[/tex] on the left side:
[tex]\[
\frac{3}{5}x - \frac{3}{5} \cdot 4 + \frac{6}{7}
= \frac{3}{5}x - \frac{12}{5} + \frac{6}{7}
\][/tex]
Distribute [tex]\(\frac{1}{7}\)[/tex] on the right side:
[tex]\[
\frac{1}{7}x + \frac{1}{7} \cdot 2
= \frac{1}{7}x + \frac{2}{7}
\][/tex]
Now we have:
[tex]\[
\frac{3}{5}x - \frac{12}{5} + \frac{6}{7} = \frac{1}{7}x + \frac{2}{7}
\][/tex]
2. Combine like terms:
To simplify, it is beneficial to get rid of the fractions by finding a common denominator. The common denominator for 5 and 7 is 35.
Multiply through by 35:
[tex]\[
35 \left( \frac{3}{5}x - \frac{12}{5} + \frac{6}{7} \right) = 35 \left( \frac{1}{7}x + \frac{2}{7} \right)
\][/tex]
Simplifying each term:
[tex]\[
35 \cdot \frac{3}{5}x = 21x, \quad 35 \cdot \frac{12}{5} = 84, \quad 35 \cdot \frac{6}{7} = 30
\][/tex]
[tex]\[
35 \cdot \frac{1}{7}x = 5x, \quad 35 \cdot \frac{2}{7} = 10
\][/tex]
So, the equation becomes:
[tex]\[
21x - 84 + 30 = 5x + 10
\][/tex]
3. Combine like terms and isolate [tex]\(x\)[/tex]:
Simplify the equation:
[tex]\[
21x - 54 = 5x + 10
\][/tex]
Subtract [tex]\(5x\)[/tex] from both sides:
[tex]\[
16x - 54 = 10
\][/tex]
Add 54 to both sides:
[tex]\[
16x = 64
\][/tex]
4. Solve for [tex]\(x\)[/tex]:
Divide both sides by 16:
[tex]\[
x = \frac{64}{16} = 4
\][/tex]
Thus, the solution to the equation is:
[tex]\[
x = 4
\][/tex]
