To solve the equation
[tex]\[
\frac{6}{t-4} = \frac{6}{3t+4}
\][/tex]
we need to find the value of [tex]\( t \)[/tex] that satisfies this equation. Let's go through the steps to solve this step-by-step.
1. Cross-Multiply to Eliminate the Fractions:
[tex]\[
6(3t + 4) = 6(t - 4)
\][/tex]
2. Distribute the Constant across the Parentheses:
[tex]\[
18t + 24 = 6t - 24
\][/tex]
3. Move All [tex]\( t \)[/tex]-Terms to One Side of the Equation:
Subtract [tex]\( 6t \)[/tex] from both sides:
[tex]\[
18t - 6t + 24 = 6t - 6t - 24
\][/tex]
Simplifies to:
[tex]\[
12t + 24 = -24
\][/tex]
4. Isolate the [tex]\( t \)[/tex]-Term:
Subtract 24 from both sides:
[tex]\[
12t + 24 - 24 = -24 - 24
\][/tex]
Simplifies to:
[tex]\[
12t = -48
\][/tex]
5. Solve for [tex]\( t \)[/tex]:
Divide both sides by 12:
[tex]\[
t = \frac{-48}{12}
\][/tex]
Which simplifies to:
[tex]\[
t = -4
\][/tex]
So the solution to the equation
[tex]\[
\frac{6}{t-4} = \frac{6}{3t+4}
\][/tex]
is:
[tex]\[
t = -4
\][/tex]
