To solve the equations given:

1. **Equations:**
\[
2x = 3y \quad (1)
\]
\[
5y = 7z \quad (2)
\]
\[
3x - 7y + 5 = -30 \quad (3)
\]

2. **From equation (1)**, we can express \(y\) in terms of \(x\):
\[
y = \frac{2}{3}x
\]

3. **From equation (2)**, we can express \(z\) in terms of \(y\):
\[
z = \frac{5}{7}y
\]

4. **Substituting \(y\) from (1) into (3)**:
\[
3x - 7\left(\frac{2}{3}x\right) + 5 = -30
\]
Simplifying:
\[
3x - \frac{14}{3}x + 5 = -30
\]
Combine like terms:
\[
\left(3 - \frac{14}{3}\right)x + 5 = -30
\]
\[
\frac{9}{3} - \frac{14}{3} = -\frac{5}{3}
\]
Thus, we have:
\[
-\frac{5}{3}x + 5 = -30
\]
Subtract 5 from both sides:
\[
-\frac{5}{3}x = -35
\]
Multiply both sides by \(-\frac{3}{5}\):
\[
x = 21
\]

5. **Finding \(y\)** using \(x = 21\):
\[
y = \frac{2}{3} \cdot 21 = 14
\]

6. **Finding \(z\)** using \(y = 14\):
\[
z = \frac{5}{7} \cdot 14 = 10
\]

7. **Final Results:**
\[
x = 21, \quad y = 14, \quad z = 10
\]