Pre-algebra worksheet using the distributive property to solve equations and decode a hidden message.

Problem Overview:

The task involves solving a series of equations that demonstrate the Distributive Property. The Distributive Property states that for any numbers \(a\), \(b\), and \(c\):

\[
a(b + c) = (a \times b) + (a \times c)
\]

We need to determine the value of each letter in the equations so that they satisfy this property. After solving, we will use the answers to decode a message using the provided code at the bottom of the page.

---

Step-by-Step Solution:

#### 1. Equation: \(-4(2 + 8) = (-4 \times 2) + (-4 \times N)\)
- Simplify the left side:
\[
-4(2 + 8) = -4 \times 10 = -40
\]
- Simplify the right side:
\[
(-4 \times 2) + (-4 \times N) = -8 + (-4N)
\]
- Set the two sides equal:
\[
-40 = -8 + (-4N)
\]
- Solve for \(N\):
\[
-40 = -8 - 4N \implies -40 + 8 = -4N \implies -32 = -4N \implies N = 8
\]

#### 2. Equation: \(15(-7 + 5) = (15 \times Y) + (15 \times 5)\)
- Simplify the left side:
\[
15(-7 + 5) = 15 \times (-2) = -30
\]
- Simplify the right side:
\[
(15 \times Y) + (15 \times 5) = 15Y + 75
\]
- Set the two sides equal:
\[
-30 = 15Y + 75
\]
- Solve for \(Y\):
\[
-30 = 15Y + 75 \implies -30 - 75 = 15Y \implies -105 = 15Y \implies Y = -7
\]

#### 3. Equation: \(-34(10 + -3) = (-34 \times 10) + (-34 \times A)\)
- Simplify the left side:
\[
-34(10 + -3) = -34 \times 7 = -238
\]
- Simplify the right side:
\[
(-34 \times 10) + (-34 \times A) = -340 + (-34A)
\]
- Set the two sides equal:
\[
-238 = -340 + (-34A)
\]
- Solve for \(A\):
\[
-238 = -340 - 34A \implies -238 + 340 = -34A \implies 102 = -34A \implies A = -3
\]

#### 4. Equation: \(7(-6 + -5) = (7 \times G) + (7 \times -5)\)
- Simplify the left side:
\[
7(-6 + -5) = 7 \times (-11) = -77
\]
- Simplify the right side:
\[
(7 \times G) + (7 \times -5) = 7G - 35
\]
- Set the two sides equal:
\[
-77 = 7G - 35
\]
- Solve for \(G\):
\[
-77 = 7G - 35 \implies -77 + 35 = 7G \implies -42 = 7G \implies G = -6
\]

#### 5. Equation: \(59(12 + 9) = (59 \times 12) + (L \times 9)\)
- Simplify the left side:
\[
59(12 + 9) = 59 \times 21 = 1239
\]
- Simplify the right side:
\[
(59 \times 12) + (L \times 9) = 708 + 9L
\]
- Set the two sides equal:
\[
1239 = 708 + 9L
\]
- Solve for \(L\):
\[
1239 = 708 + 9L \implies 1239 - 708 = 9L \implies 531 = 9L \implies L = 59
\]

#### 6. Equation: \(20(-8 + 1) = (C \times -8) + (20 \times 1)\)
- Simplify the left side:
\[
20(-8 + 1) = 20 \times (-7) = -140
\]
- Simplify the right side:
\[
(C \times -8) + (20 \times 1) = -8C + 20
\]
- Set the two sides equal:
\[
-140 = -8C + 20
\]
- Solve for \(C\):
\[
-140 = -8C + 20 \implies -140 - 20 = -8C \implies -160 = -8C \implies C = 20
\]

#### 7. Equation: \(-17(-4 + -5) = (-17 \times -4) + (S \times -5)\)
- Simplify the left side:
\[
-17(-4 + -5) = -17 \times (-9) = 153
\]
- Simplify the right side:
\[
(-17 \times -4) + (S \times -5) = 68 + (-5S)
\]
- Set the two sides equal:
\[
153 = 68 + (-5S)
\]
- Solve for \(S\):
\[
153 = 68 - 5S \implies 153 - 68 = -5S \implies 85 = -5S \implies S = -17
\]

#### 8. Equation: \((6 \times 43) + (6 \times 19) = 6(43 + H)\)
- Simplify the left side:
\[
(6 \times 43) + (6 \times 19) = 258 + 114 = 372
\]
- Simplify the right side:
\[
6(43 + H) = 6 \times (43 + H)
\]
- Set the two sides equal:
\[
372 = 6(43 + H)
\]
- Solve for \(H\):
\[
372 = 6(43 + H) \implies 372 = 258 + 6H \implies 372 - 258 = 6H \implies 114 = 6H \implies H = 19
\]

#### 9. Equation: \((-7 \times 28) + (-7 \times 75) = -7(28 + R)\)
- Simplify the left side:
\[
(-7 \times 28) + (-7 \times 75) = -196 + (-525) = -721
\]
- Simplify the right side:
\[
-7(28 + R) = -7 \times (28 + R)
\]
- Set the two sides equal:
\[
-721 = -7(28 + R)
\]
- Solve for \(R\):
\[
-721 = -7(28 + R) \implies -721 = -196 - 7R \implies -721 + 196 = -7R \implies -525 = -7R \implies R = 75
\]

#### 10. Equation: \((8 \times -4) + (8 \times 18) = 8(I + 18)\)
- Simplify the left side:
\[
(8 \times -4) + (8 \times 18) = -32 + 144 = 112
\]
- Simplify the right side:
\[
8(I + 18) = 8I + 144
\]
- Set the two sides equal:
\[
112 = 8I + 144
\]
- Solve for \(I\):
\[
112 = 8I + 144 \implies 112 - 144 = 8I \implies -32 = 8I \implies I = -4
\]

#### 11. Equation: \((-6 \times 13) + (-6 \times 30) = -6(W + 30)\)
- Simplify the left side:
\[
(-6 \times 13) + (-6 \times 30) = -78 + (-180) = -258
\]
- Simplify the right side:
\[
-6(W + 30) = -6W - 180
\]
- Set the two sides equal:
\[
-258 = -6W - 180
\]
- Solve for \(W\):
\[
-258 = -6W - 180 \implies -258 + 180 = -6W \implies -78 = -6W \implies W = 13
\]

#### 12. Equation: \((-19 \times -8) + (-19 \times 52) = P(-8 + 52)\)
- Simplify the left side:
\[
(-19 \times -8) + (-19 \times 52) = 152 + (-988) = -836
\]
- Simplify the right side:
\[
P(-8 + 52) = P \times 44
\]
- Set the two sides equal:
\[
-836 = P \times 44
\]
- Solve for \(P\):
\[
-836 = 44P \implies P = -19
\]

#### 13. Equation: \((98 \times -2) + (98 \times -11) = E(-2 + -11)\)
- Simplify the left side:
\[
(98 \times -2) + (98 \times -11) = -196 + (-1078) = -1274
\]
- Simplify the right side:
\[
E(-2 + -11) = E \times (-13)
\]
- Set the two sides equal:
\[
-1274 = E \times (-13)
\]
- Solve for \(E\):
\[
-1274 = -13E \implies E = 98
\]

#### 14. Equation: \((50 \times 6) + (-31 \times 50) = T(6 + -31)\)
- Simplify the left side:
\[
(50 \times 6) + (-31 \times 50) = 300 + (-1550) = -1250
\]
- Simplify the right side:
\[
T(6 + -31) = T \times (-25)
\]
- Set the two sides equal:
\[
-1250 = T \times (-25)
\]
- Solve for \(T\):
\[
-1250 = -25T \implies T = 50
\]

---

Final Answers:

\[
\begin{aligned}
&N = 8, \quad Y = -7, \quad A = -3, \quad G = -6, \quad L = 59, \\
&C = 20, \quad S = -17, \quad H = 19, \quad R = 75, \quad I = -4, \\
&W = 13, \quad P = -19, \quad E = 98, \quad T = 50.
\end{aligned}
\]

---

Decoding the Message:

Using the code at the bottom of the page, we replace each number with its corresponding letter:

\[
\begin{aligned}
&98 \to E, \quad 59 \to L, \quad 98 \to E, \quad 20 \to C, \quad 50 \to T, \quad 75 \to R, \\
&-4 \to I, \quad 20 \to C, \quad -4 \to I, \quad 50 \to T, \quad -7 \to Y, \quad -4 \to I, \\
&-17 \to S.
\end{aligned}
\]

The decoded message is: "ELECTRICITY".

---

Final Answer:

\[
\boxed{ELECTRICITY}
\]