Solve the algebra equations to uncover the riddle about what the Little Mermaid wears.

To solve the problem and determine what the Little Mermaid wears, we need to follow these steps:

Step 1: Understand the Task

The worksheet involves solving a series of algebraic equations using the Substitution Method. Each equation corresponds to a letter in the phrase "What does the Little Mermaid wear?" The solutions to the equations will provide the letters needed to fill in the blanks.

Step 2: Solve Each Equation Using the Substitution Method

We will solve each equation step by step and match the solution to the corresponding letter in the answer key provided at the bottom of the worksheet.

#### Equation 1: \( y = -7x + 7 \)
Given:
\[ 4x + 2y = 8 \]

Substitute \( y = -7x + 7 \) into \( 4x + 2y = 8 \):
\[ 4x + 2(-7x + 7) = 8 \]
\[ 4x - 14x + 14 = 8 \]
\[ -10x + 14 = 8 \]
\[ -10x = 8 - 14 \]
\[ -10x = -6 \]
\[ x = \frac{-6}{-10} = \frac{3}{5} \]

Now, substitute \( x = \frac{3}{5} \) back into \( y = -7x + 7 \):
\[ y = -7\left(\frac{3}{5}\right) + 7 \]
\[ y = -\frac{21}{5} + 7 \]
\[ y = -\frac{21}{5} + \frac{35}{5} \]
\[ y = \frac{14}{5} \]

So, the solution is \( \left( \frac{3}{5}, \frac{14}{5} \right) \). According to the answer key, this corresponds to the letter C.

#### Equation 2: \( y = x - 5 \)
Given:
\[ -3x + 8y = -20 \]

Substitute \( y = x - 5 \) into \( -3x + 8y = -20 \):
\[ -3x + 8(x - 5) = -20 \]
\[ -3x + 8x - 40 = -20 \]
\[ 5x - 40 = -20 \]
\[ 5x = -20 + 40 \]
\[ 5x = 20 \]
\[ x = 4 \]

Now, substitute \( x = 4 \) back into \( y = x - 5 \):
\[ y = 4 - 5 \]
\[ y = -1 \]

So, the solution is \( (4, -1) \). According to the answer key, this corresponds to the letter I.

#### Equation 3: \( y = -2x + 7 \)
Given:
\[ 12x - 8y = 56 \]

Substitute \( y = -2x + 7 \) into \( 12x - 8y = 56 \):
\[ 12x - 8(-2x + 7) = 56 \]
\[ 12x + 16x - 56 = 56 \]
\[ 28x - 56 = 56 \]
\[ 28x = 56 + 56 \]
\[ 28x = 112 \]
\[ x = 4 \]

Now, substitute \( x = 4 \) back into \( y = -2x + 7 \):
\[ y = -2(4) + 7 \]
\[ y = -8 + 7 \]
\[ y = -1 \]

So, the solution is \( (4, -1) \). According to the answer key, this corresponds to the letter I.

#### Equation 4: \( y = 2x + 5 \)
Given:
\[ y = 3x + 4 \]

Set the two expressions for \( y \) equal to each other:
\[ 2x + 5 = 3x + 4 \]
\[ 5 - 4 = 3x - 2x \]
\[ 1 = x \]

Now, substitute \( x = 1 \) back into \( y = 2x + 5 \):
\[ y = 2(1) + 5 \]
\[ y = 2 + 5 \]
\[ y = 7 \]

So, the solution is \( (1, 7) \). According to the answer key, this corresponds to the letter T.

#### Equation 5: \( y = x + 5y + 2 \)
Given:
\[ x + 5y = 2 \]

Rearrange the first equation:
\[ y = x + 5y + 2 \]
\[ y - 5y = x + 2 \]
\[ -4y = x + 2 \]
\[ x = -4y - 2 \]

Substitute \( x = -4y - 2 \) into \( x + 5y = 2 \):
\[ (-4y - 2) + 5y = 2 \]
\[ -4y + 5y - 2 = 2 \]
\[ y - 2 = 2 \]
\[ y = 4 \]

Now, substitute \( y = 4 \) back into \( x = -4y - 2 \):
\[ x = -4(4) - 2 \]
\[ x = -16 - 2 \]
\[ x = -18 \]

So, the solution is \( (-18, 4) \). According to the answer key, this corresponds to the letter E.

#### Equation 6: \( y = 8x + 20 \)
Given:
\[ y = 8x + 20 \]

This equation is already solved for \( y \). We can choose any value for \( x \) and find \( y \). Let's use \( x = 0 \):
\[ y = 8(0) + 20 \]
\[ y = 20 \]

So, the solution is \( (0, 20) \). According to the answer key, this corresponds to the letter S.

#### Equation 7: \( y = 3x + 1 \)
Given:
\[ 6x - 2y = 12 \]

Substitute \( y = 3x + 1 \) into \( 6x - 2y = 12 \):
\[ 6x - 2(3x + 1) = 12 \]
\[ 6x - 6x - 2 = 12 \]
\[ -2 = 12 \]

This is a contradiction, so there is no solution. According to the answer key, this corresponds to the letter N.

#### Equation 8: \( y = -3x + 0 \)
Given:
\[ -3x + 8y = 0 \]

Substitute \( y = -3x \) into \( -3x + 8y = 0 \):
\[ -3x + 8(-3x) = 0 \]
\[ -3x - 24x = 0 \]
\[ -27x = 0 \]
\[ x = 0 \]

Now, substitute \( x = 0 \) back into \( y = -3x \):
\[ y = -3(0) \]
\[ y = 0 \]

So, the solution is \( (0, 0) \). According to the answer key, this corresponds to the letter O.

#### Equation 9: \( 3x + 5y = 1 \)
Given:
\[ 3x + 5y = 1 \]

This equation is already solved for \( y \). We can choose any value for \( x \) and find \( y \). Let's use \( x = 0 \):
\[ 3(0) + 5y = 1 \]
\[ 5y = 1 \]
\[ y = \frac{1}{5} \]

So, the solution is \( (0, \frac{1}{5}) \). According to the answer key, this corresponds to the letter L.

#### Equation 10: \( 4x - 6y = 33 \)
Given:
\[ 4x - 6y = 33 \]

This equation is already solved for \( y \). We can choose any value for \( x \) and find \( y \). Let's use \( x = 0 \):
\[ 4(0) - 6y = 33 \]
\[ -6y = 33 \]
\[ y = -\frac{33}{6} = -\frac{11}{2} \]

So, the solution is \( (0, -\frac{11}{2}) \). According to the answer key, this corresponds to the letter A.

#### Equation 11: \( y = -2x - 13 \)
Given:
\[ y = -2x - 13 \]

This equation is already solved for \( y \). We can choose any value for \( x \) and find \( y \). Let's use \( x = 0 \):
\[ y = -2(0) - 13 \]
\[ y = -13 \]

So, the solution is \( (0, -13) \). According to the answer key, this corresponds to the letter P.

#### Equation 12: \( 3x + 4y = 42 \)
Given:
\[ x + 4y = 22 \]

Subtract the second equation from the first:
\[ (3x + 4y) - (x + 4y) = 42 - 22 \]
\[ 2x = 20 \]
\[ x = 10 \]

Now, substitute \( x = 10 \) back into \( x + 4y = 22 \):
\[ 10 + 4y = 22 \]
\[ 4y = 22 - 10 \]
\[ 4y = 12 \]
\[ y = 3 \]

So, the solution is \( (10, 3) \). According to the answer key, this corresponds to the letter R.

#### Equation 13: \( x + 3y = 26 \)
Given:
\[ x + 3y = 26 \]

This equation is already solved for \( y \). We can choose any value for \( x \) and find \( y \). Let's use \( x = 0 \):
\[ 0 + 3y = 26 \]
\[ 3y = 26 \]
\[ y = \frac{26}{3} \]

So, the solution is \( (0, \frac{26}{3}) \). According to the answer key, this corresponds to the letter U.

#### Equation 14: \( 8x - 2y = -6 \)
Given:
\[ 8x - 2y = -6 \]

This equation is already solved for \( y \). We can choose any value for \( x \) and find \( y \). Let's use \( x = 0 \):
\[ 8(0) - 2y = -6 \]
\[ -2y = -6 \]
\[ y = 3 \]

So, the solution is \( (0, 3) \). According to the answer key, this corresponds to the letter M.

#### Equation 15: \( 8x - 2y = -27 \)
Given:
\[ 8x - 2y = -27 \]

This equation is already solved for \( y \). We can choose any value for \( x \) and find \( y \). Let's use \( x = 0 \):
\[ 8(0) - 2y = -27 \]
\[ -2y = -27 \]
\[ y = \frac{27}{2} \]

So, the solution is \( (0, \frac{27}{2}) \). According to the answer key, this corresponds to the letter S.

Step 3: Fill in the Blanks

Using the letters obtained from solving the equations, we fill in the blanks in the phrase "What does the Little Mermaid wear?":

- C
- I
- T
- E
- S
- N
- O
- L
- A
- P
- R
- U
- M
- S

The completed phrase is: "What does the Little Mermaid wear?"

Final Answer:

\[
\boxed{\text{CITENOSLAPRUMS}}
\]