$$\frac{9t-7}{3t+5}=\frac{3t-4}{t+6}$$
$t=\frac{1}{2}=0.5$
$$\left(t+6\right)\left(9t-7\right)=\left(3t+5\right)\left(3t-4\right)$$
$$9t^{2}+47t-42=\left(3t+5\right)\left(3t-4\right)$$
$$9t^{2}+47t-42=9t^{2}+3t-20$$
$$9t^{2}+47t-42-9t^{2}=3t-20$$
$$47t-42=3t-20$$
$$47t-42-3t=-20$$
$$44t-42=-20$$
$$44t=-20+42$$
$$44t=22$$
$$t=\frac{22}{44}$$
$$t=\frac{1}{2}$$
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