https://r2.erweima.ai/mathbot/image/1726374554442bmwl537s.jpg

Let's solve the expressions one by one from the image you uploaded.

1. \(\frac{3}{s - t}\) at \(s = 2\), \(t = -6\):
   \[
   \frac{3}{s - t} = \frac{3}{2 - (-6)} = \frac{3}{2 + 6} = \frac{3}{8}
   \]

2. \(\frac{x + 1}{y^2}\) at \(x = 8\), \(y = \frac{3}{5}\):
   \[
   \frac{8 + 1}{\left(\frac{3}{5}\right)^2} = \frac{9}{\frac{9}{25}} = 9 \times \frac{25}{9} = 25
   \]

3. \(\frac{3}{u} (u^2 + 68)\) at \(u = -12\):
   \[
   \frac{3}{-12}((-12)^2 + 68) = -\frac{1}{4} (144 + 68) = -\frac{1}{4} \times 212 = -53
   \]

4. \(21r(63r + 3)\) at \(r = -\frac{3}{7}\):
   \[
   21\left(-\frac{3}{7}\right)(63\left(-\frac{3}{7}\right) + 3) = 21\left(-\frac{3}{7}\right)\left(-27 + 3\right) = 21\left(-\frac{3}{7}\right)(-24) = 216
   \]

5. \(6n^2 + 4n^2 - 19\) at \(n = -3\):
   \[
   6(-3)^2 + 4(-3)^2 - 19 = 6(9) + 4(9) - 19 = 54 + 36 - 19 = 71
   \]

6. \(\frac{12q - 34}{4}\) at \(q = -\frac{1}{3}\):
   \[
   \frac{12\left(-\frac{1}{3}\right) - 34}{4} = \frac{-4 - 34}{4} = \frac{-38}{4} = -9.5
   \]

7. \(\frac{5t}{3t - s} - 1\) at \(s = 2\), \(t = -6\):
   \[
   \frac{5(-6)}{3(-6) - 2} - 1 = \frac{-30}{-18 - 2} - 1 = \frac{-30}{-20} - 1 = 1.5 - 1 = 0.5
   \]

8. \(\frac{s - 4}{t^2}\) at \(s = 5\), \(t = -5\):
   \[
   \frac{5 - 4}{(-5)^2} = \frac{1}{25}
   \]

9. \(\frac{6q - g}{4}\) at \(q = -4\), \(g = 1\):
   \[
   \frac{6(-4) - 1}{4} = \frac{-24 - 1}{4} = \frac{-25}{4} = -6.25
   \]

10. \((10p - 4)^2\) at \(p = 2\):
   \[
   (10(2) - 4)^2 = (20 - 4)^2 = 16^2 = 256
   \]

If you'd like, I can provide more details or clarifications! Here are some related questions:

1. What happens if \(t\) and \(s\) values are swapped in the equations?
2. How would the result change if we used fractional values for all variables?
3. How can we simplify expressions involving squares and fractions?
4. Can we solve these using an alternate method like factoring?
5. What happens when values lead to negative or zero in the denominators?

**Tip:** Always check for cases where the denominator could become zero to avoid undefined expressions!

Evaluate the algebraic expressions for the given values of each variable.

Step-by-step solutions for evaluating algebraic expressions by substituting given values into variables. This includes simplifying fractions and quadratic expressions, covering essential algebra concepts for Grades 7-9.

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