(c) $$ y+5 x=-21 $$ $$ \frac{10}{9 x-7 y}=\frac{2}{y+x} $$

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To solve the system of equations:

$$
\begin{cases}
y + 5x = -21 \quad \text{(1)} \
\frac{10}{9x - 7y} = \frac{2}{x + y} \quad \text{(2)}
\end{cases}
$$

**Step 1: Express $$ y $$ in terms of $$ x $$ from equation (1).**

$$
y = -21 - 5x
$$

**Step 2: Substitute $$ y = -21 - 5x $$ into equation (2).**

$$
\frac{10}{9x - 7(-21 - 5x)} = \frac{2}{x + (-21 - 5x)}
$$

Simplify the denominators:

$$
9x + 147 + 35x = 44x + 147
$$

$$
x - 5x - 21 = -4x - 21
$$

So the equation becomes:

$$
\frac{10}{44x + 147} = \frac{2}{-4x - 21}
$$

**Step 3: Cross-multiply to eliminate the fractions.**

$$
10(-4x - 21) = 2(44x + 147)
$$

$$
-40x - 210 = 88x + 294
$$

**Step 4: Solve for $$ x $$.**

$$
-40x - 88x = 294 + 210
$$

$$
-128x = 504
$$

$$
x = \frac{504}{-128} = -\frac{63}{16}
$$

**Step 5: Substitute $$ x = -\frac{63}{16} $$ back into equation (1) to find $$ y $$.**

$$
y = -21 - 5\left(-\frac{63}{16}\right) = -21 + \frac{315}{16}
$$

$$
y = -\frac{336}{16} + \frac{315}{16} = -\frac{21}{16}
$$

**Final Solution:**

$$
x = -\frac{63}{16}, \quad y = -\frac{21}{16}
$$

**Verification:**

1. Substitute $$ x $$ and $$ y $$ into equation (1):

$$
-\frac{21}{16} + 5\left(-\frac{63}{16}\right) = -\frac{21}{16} - \frac{315}{16} = -\frac{336}{16} = -21
$$

2. Substitute $$ x $$ and $$ y $$ into equation (2):

$$
\frac{10}{9\left(-\frac{63}{16}\right) - 7\left(-\frac{21}{16}\right)} = \frac{10}{-\frac{567}{16} + \frac{147}{16}} = \frac{10}{-\frac{420}{16}} = \frac{10}{-\frac{105}{4}} = -\frac{40}{105} = -\frac{8}{21}
$$

$$
\frac{2}{-\frac{63}{16} + \left(-\frac{21}{16}\right)} = \frac{2}{-\frac{84}{16}} = \frac{2}{-\frac{21}{4}} = -\frac{8}{21}
$$

Both sides of equation (2) are equal, confirming the solution is correct.

**Answer:**
The system has the solution $$ x = -\,\dfrac{63}{16} $$ and $$ y = -\,\dfrac{21}{16} $$.
The solution to the system of equations is $$ x = -\frac{63}{16} $$ and $$ y = -\frac{21}{16} $$.

(c) \( y+5 x=-21 \) \( \frac{10}{9 x-7 y}=\frac{2}{y+x} \)

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