Problem 13

Multiplying both sides of the equation by $2 \cdot 3 = 6$, we obtain $6 $\cdot \dfrac{2t}{3} + 6\cdot\dfrac{3t}{2}$ = 6
$\cdot$ 26$or$4t + 9t =
156$.

Simplifying, we obtain $13t = 156$
and so $t = 12$.

Alternatively, using a common denominator of $2 \cdot 3 = 6$, we obtain $\text{2}$ 2t}{2 $\cdot$ 3} + $\text{3}$
3t}{3 $\cdot$ 2} = 26$or$$\dfrac{4t}{6} + \dfrac{9t}{6}$ = 26$.

Simplifying, we obtain $\$\dfrac{13t}{6}$ =
26$and so$13t = 6 $\cdot$ 26$
or $t = 6 \cdot 2 = 12$.
Multiplying both sides of the equation by $8$, we obtain $\dfrac{8(3+x)}{4} = 6 + x$ which
simplifies to $2(3+x) = 6 + x$.

Simplifying further, we obtain $6 + 2x = 6 +
x$and so$x = 0$.

Alternatively, splitting each fraction into two pieces, we obtain
$\$\dfrac{3}{4} + \dfrac{x}{4} = \dfrac{6}{8} + \dfrac{x}{8}\$.$

Since $\dfrac{3}{4} = \dfrac{6}{8}$,
we obtain $\$\dfrac{x}{4} = \dfrac{x}{8}$$and so$8x =
4x$or$x = 0$.
Since $y > 0$, then $\sqrt{y^2} = y$.

From the given equation, we obtain $\$\sqrt{9}$ +
16 + 144} = $\sqrt{9}$ + 16} + y$.

Thus, $\sqrt{169} = \sqrt{25} + y$ or
$13 = 5 + y$ and so $y = 8$.