To solve the rational equation [tex]\(\frac{4}{t} - \frac{8}{7} = -\frac{4}{t}\)[/tex], follow these steps:
1. Combine the terms involving [tex]\(t\)[/tex] on one side of the equation:
[tex]\[
\frac{4}{t} - \frac{8}{7} = -\frac{4}{t}
\][/tex]
Move [tex]\(\frac{4}{t}\)[/tex] from the left-hand side to the right-hand side by adding [tex]\(\frac{4}{t}\)[/tex] to both sides:
[tex]\[
\frac{4}{t} + \frac{4}{t} = \frac{8}{7}
\][/tex]
2. Simplify the left-hand side:
[tex]\[
\frac{4}{t} + \frac{4}{t} = \frac{8}{t}
\][/tex]
Combining the fractions:
[tex]\[
\frac{8}{t} = \frac{8}{7}
\][/tex]
3. Solve for [tex]\(t\)[/tex]:
To isolate [tex]\(t\)[/tex], cross-multiply:
[tex]\[
8 \cdot 7 = 8 \cdot t
\][/tex]
Simplify the equation:
[tex]\[
56 = 8t
\][/tex]
Divide both sides by 8 to solve for [tex]\(t\)[/tex]:
[tex]\[
t = \frac{56}{8}
\][/tex]
4. Simplify the fraction:
[tex]\[
t = 7
\][/tex]
Therefore, the solution to the equation is:
[tex]\[
t = 7
\][/tex]
So, [tex]\(t = 7\)[/tex].
