Solve for $$ y $$ in each of the following. a. $$ \frac{2}{3} y=\frac{2}{7} $$ b. $$ \frac{6}{7} \div y=\frac{1}{6} $$. c. $$ \frac{13}{14}-\frac{6}{7} y=\frac{7}{8} $$ d. $$ \frac{2 y}{3}+\frac{1}{4}=\frac{y}{6}-\frac{1}{2} $$ $$ y=|\overline{7}| $$ (Jmpuly your answer. rype an mineger, iracuon, of mixea mumber.) c. $$ \frac{13}{14}-\frac{6}{7} y=\frac{7}{8} $$ $$ y=\frac{1}{16} $$ (Simplify your answer. Type an integer, fraction, or mixed number.) d. $$ \frac{2 y}{3}+\frac{1}{4}=\frac{y}{6}-\frac{1}{2} $$ $$ y=\square $$ (Simplify your answer. Type an integer, fraction, or mixed number.)

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UpStudy AI · Accepted Answer

**a. Solve $$\frac{2}{3}y=\frac{2}{7}$$:**

Multiply both sides by $$\frac{3}{2}$$ to isolate $$y$$:
$$
y = \frac{2}{7} \times \frac{3}{2} = \frac{6}{14} = \frac{3}{7}.
$$

---

**b. Solve $$\frac{6}{7} \div y=\frac{1}{6}$$:**

Interpret the division as:
$$
\frac{6}{7} \times \frac{1}{y} = \frac{1}{6}.
$$
This can be written as:
$$
\frac{6}{7y} = \frac{1}{6}.
$$

Multiply both sides by $$7y$$ to clear the fraction:
$$
6 = \frac{7y}{6}.
$$

Then multiply both sides by 6:
$$
36 = 7y.
$$

Solve for $$y$$ by dividing by 7:
$$
y = \frac{36}{7}.
$$

---

**c. Solve $$\frac{13}{14} - \frac{6}{7}y = \frac{7}{8}$$:**

Isolate the term containing $$y$$ by subtracting $$\frac{13}{14}$$ from both sides:
$$
-\frac{6}{7}y = \frac{7}{8} - \frac{13}{14}.
$$

Find a common denominator for the right-hand side. The least common denominator for 8 and 14 is 56:
$$
\frac{7}{8} = \frac{49}{56}, \quad \frac{13}{14} = \frac{52}{56}.
$$
Thus,
$$
-\frac{6}{7}y = \frac{49}{56} - \frac{52}{56} = -\frac{3}{56}.
$$

Multiply both sides by $$-1$$:
$$
\frac{6}{7}y = \frac{3}{56}.
$$

Now, solve for $$y$$ by multiplying both sides by the reciprocal $$\frac{7}{6}$$:
$$
y = \frac{3}{56} \times \frac{7}{6} = \frac{21}{336}.
$$

Simplify the fraction by dividing numerator and denominator by 21:
$$
y = \frac{1}{16}.
$$

---

**d. Solve $$\frac{2y}{3} + \frac{1}{4} = \frac{y}{6} - \frac{1}{2}$$:**

Clear the fractions by multiplying every term by the least common multiple of the denominators $$3$$, $$4$$, $$6$$, and $$2$$. The least common multiple is $$12$$:
$$
12 \times \frac{2y}{3} + 12 \times \frac{1}{4} = 12 \times \frac{y}{6} - 12 \times \frac{1}{2}.
$$

Compute each term:
$$
12 \times \frac{2y}{3} = 8y, \quad 12 \times \frac{1}{4} = 3,
$$
$$
12 \times \frac{y}{6} = 2y, \quad 12 \times \frac{1}{2} = 6.
$$

The equation becomes:
$$
8y + 3 = 2y - 6.
$$

Subtract $$2y$$ from both sides:
$$
6y + 3 = -6.
$$

Subtract 3 from both sides:
$$
6y = -9.
$$

Divide by 6:
$$
y = -\frac{9}{6} = -\frac{3}{2}.
$$

---

**Final Answers:**
- a. $$\displaystyle y=\frac{3}{7}$$
- b. $$\displaystyle y=\frac{36}{7}$$
- c. $$\displaystyle y=\frac{1}{16}$$
- d. $$\displaystyle y=-\frac{3}{2}$$
a. $$ y = \frac{3}{7} $$ b. $$ y = \frac{36}{7} $$ c. $$ y = \frac{1}{16} $$ d. $$ y = -\frac{3}{2} $$

Solve for in each of the following.
a.
b. .
c.
d.
(Jmpuly your answer. rype an mineger, iracuon, of mixea mumber.)
c.
(Simplify your answer. Type an integer, fraction, or mixed number.)
d.
(Simplify your answer. Type an integer, fraction, or mixed number.)

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Solve for in each of the following. a. b. . c. d. (Jmpuly your answer. rype an mineger, iracuon, of mixea mumber.) c. (Simplify your answer. Type an integer, fraction, or mixed number.) d. (Simplify your answer. Type an integer, fraction, or mixed number.)