Let's go through the math problems one by one. The sheet asks to "Evaluate each using the values given." I'll break down each problem, showing how to substitute the given values and simplify the expressions.

1) $a - (4 - 7 + b + c)$

Given values: $a = 10$, $b = 3$, $c = 1$

Substituting the values: $10 - (4 - 7 + 3 + 1) = 10 - (4 - 7 + 3 + 1) = 10 - (1) = 9$ Answer: $9$

2) $p - p + \frac{q}{2}$

Given values: $p = 8$, $q = 2$

Substituting the values: $8 - 8 + \frac{2}{2} = 0 + 1 = 1$ Answer: $1$

3) $-6 - ac(b + \frac{7}{c})$

Given values: $a = -6$, $b = -7$, $c = 5$

Substituting the values: $-6 - (-6)(5)(-7 + \frac{7}{5}) = -6 - (-6)(5)(-7 + 1.4) = -6 - (-6)(5)(-5.6)$ $= -6 + 6(5)(5.6) = -6 + 168 = 162$ Answer: $162$

4) $r - p^2 + q^2$

Given values: $r = 1$, $p = -2$, $q = 6$

Substituting the values: $1 - (-2)^2 + 6^2 = 1 - 4 + 36 = 33$ Answer: $33$

5) $|z| (y + z)^3$

Given values: $y = -5$, $z = 3$

Substituting the values: $|3| (-5 + 3)^3 = 3 \times (-2)^3 = 3 \times (-8) = -24$ Answer: $-24$

6) $x^2 - (z - 7^2)$

Given values: $x = 3$, $z = -10$

Substituting the values: $3^2 - (-10 - 7^2) = 9 - (-10 - 49) = 9 - (-59) = 9 + 59 = 68$ Answer: $68$

7) $y + x \left(\frac{8}{4}\right)^2$

Given values: $x = -8$, $y = -3$

Substituting the values: $-3 + (-8) \left(\frac{8}{4}\right)^2 = -3 + (-8)(2^2) = -3 + (-8)(4) = -3 - 32 = -35$ Answer: $-35$

8) $y - \left( x - 8 + \frac{z}{2} \right)$

Given values: $x = -10$, $y = -7$, $z = -2$

Substituting the values: $-7 - \left( -10 - 8 + \frac{-2}{2} \right) = -7 - \left( -10 - 8 - 1 \right) = -7 - (-19) = -7 + 19 = 12$ Answer: $12$

Solve Each Equation

9) $5 = k + 7$

Subtract 7 from both sides: $k = 5 - 7 = -2$ Answer: $k = -2$

10) $6 = x - (-4)$

Simplify the double negative and solve: $6 = x + 4 \quad \Rightarrow \quad x = 6 - 4 = 2$ Answer: $x = 2$

11) $-65 = -5k$

Divide both sides by -5: $k = \frac{-65}{-5} = 13$ Answer: $k = 13$

12) $-18 + n = -38$

Add 18 to both sides: $n = -38 + 18 = -20$ Answer: $n = -20$

Let me know if you'd like further details or explanations! Here are some additional related questions:

1. Can you simplify algebraic expressions with multiple variables?
2. How do you solve quadratic equations with two unknowns?
3. What strategies can help in solving absolute value equations?
4. How do you approach solving systems of equations?
5. What is the process for solving equations with fractions?

Tip: When substituting values into equations, always handle negative numbers with extra care to avoid sign errors.