Question

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distributive property 2 - name: \underline{\quad\quad\quad\quad\quad} class: \underline{\quad\quad\quad\quad\quad}
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$$\begin{tabular}{|c|c|c|c|c|} \\hline q & expressions & what i spent & simplified form \\\\ \\hline 1 & $-4 + 3(2c + 5) - c$ & & \\\\ \\hline 2 & $2(7n - 4) + 8 - 5n$ & & \\\\ \\hline 3 & $10 - (4x - 6) + 3(2x + 1)$ & & \\\\ \\hline 4 & $2(3m - 2) - (m + 6)$ & & \\\\ \\hline \\end{tabular}$$

Explanation:

To solve these algebraic expressions using the distributive property, we'll simplify each one step by step.

Q1: \( -4 + 3(2c + 5) - c \)

Step 1: Distribute the 3

Multiply 3 with each term inside the parentheses:
\( 3(2c + 5) = 3 \cdot 2c + 3 \cdot 5 = 6c + 15 \)

Step 2: Substitute back and combine like terms

Substitute \( 6c + 15 \) into the expression:
\( -4 + 6c + 15 - c \)

Combine the constant terms (\( -4 + 15 \)) and the \( c \)-terms (\( 6c - c \)):
\( (-4 + 15) + (6c - c) = 11 + 5c \)

Q2: \( 2(7n - 4) + 8 - 5n \)

Step 1: Distribute the 2

Multiply 2 with each term inside the parentheses:
\( 2(7n - 4) = 2 \cdot 7n - 2 \cdot 4 = 14n - 8 \)

Step 2: Substitute back and combine like terms

Substitute \( 14n - 8 \) into the expression:
\( 14n - 8 + 8 - 5n \)

Combine the constant terms (\( -8 + 8 \)) and the \( n \)-terms (\( 14n - 5n \)):
\( (14n - 5n) + (-8 + 8) = 9n + 0 = 9n \)

Q3: \( 10 - (4x - 6) + 3(2x + 1) \)

Step 1: Distribute the negative sign and the 3

• Distribute the negative sign: \( -(4x - 6) = -4x + 6 \)
• Distribute the 3: \( 3(2x + 1) = 3 \cdot 2x + 3 \cdot 1 = 6x + 3 \)

Step 2: Substitute back and combine like terms

Substitute \( -4x + 6 \) and \( 6x + 3 \) into the expression:
\( 10 - 4x + 6 + 6x + 3 \)

Combine the constant terms (\( 10 + 6 + 3 \)) and the \( x \)-terms (\( -4x + 6x \)):
\( (10 + 6 + 3) + (-4x + 6x) = 19 + 2x \)

Q4: \( 2(3m - 2) - (m + 6) \)

Step 1: Distribute the 2 and the negative sign

• Distribute the 2: \( 2(3m - 2) = 2 \cdot 3m - 2 \cdot 2 = 6m - 4 \)
• Distribute the negative sign: \( -(m + 6) = -m - 6 \)

Step 2: Substitute back and combine like terms

Substitute \( 6m - 4 \) and \( -m - 6 \) into the expression:
\( 6m - 4 - m - 6 \)

Combine the \( m \)-terms (\( 6m - m \)) and the constant terms (\( -4 - 6 \)):
\( (6m - m) + (-4 - 6) = 5m - 10 \)

Final Answers

1. \( \boldsymbol{11 + 5c} \)
2. \( \boldsymbol{9n} \)
3. \( \boldsymbol{19 + 2x} \)
4. \( \boldsymbol{5m - 10} \)

Answer:

To solve these algebraic expressions using the distributive property, we'll simplify each one step by step.

Q1: \( -4 + 3(2c + 5) - c \)

Step 1: Distribute the 3

Multiply 3 with each term inside the parentheses:
\( 3(2c + 5) = 3 \cdot 2c + 3 \cdot 5 = 6c + 15 \)

Step 2: Substitute back and combine like terms

Substitute \( 6c + 15 \) into the expression:
\( -4 + 6c + 15 - c \)

Combine the constant terms (\( -4 + 15 \)) and the \( c \)-terms (\( 6c - c \)):
\( (-4 + 15) + (6c - c) = 11 + 5c \)

Q2: \( 2(7n - 4) + 8 - 5n \)

Step 1: Distribute the 2

Multiply 2 with each term inside the parentheses:
\( 2(7n - 4) = 2 \cdot 7n - 2 \cdot 4 = 14n - 8 \)

Step 2: Substitute back and combine like terms

Substitute \( 14n - 8 \) into the expression:
\( 14n - 8 + 8 - 5n \)

Combine the constant terms (\( -8 + 8 \)) and the \( n \)-terms (\( 14n - 5n \)):
\( (14n - 5n) + (-8 + 8) = 9n + 0 = 9n \)

Q3: \( 10 - (4x - 6) + 3(2x + 1) \)

Step 1: Distribute the negative sign and the 3

• Distribute the negative sign: \( -(4x - 6) = -4x + 6 \)
• Distribute the 3: \( 3(2x + 1) = 3 \cdot 2x + 3 \cdot 1 = 6x + 3 \)

Step 2: Substitute back and combine like terms

Substitute \( -4x + 6 \) and \( 6x + 3 \) into the expression:
\( 10 - 4x + 6 + 6x + 3 \)

Combine the constant terms (\( 10 + 6 + 3 \)) and the \( x \)-terms (\( -4x + 6x \)):
\( (10 + 6 + 3) + (-4x + 6x) = 19 + 2x \)

Q4: \( 2(3m - 2) - (m + 6) \)

Step 1: Distribute the 2 and the negative sign

• Distribute the 2: \( 2(3m - 2) = 2 \cdot 3m - 2 \cdot 2 = 6m - 4 \)
• Distribute the negative sign: \( -(m + 6) = -m - 6 \)

Step 2: Substitute back and combine like terms

Substitute \( 6m - 4 \) and \( -m - 6 \) into the expression:
\( 6m - 4 - m - 6 \)

Combine the \( m \)-terms (\( 6m - m \)) and the constant terms (\( -4 - 6 \)):
\( (6m - m) + (-4 - 6) = 5m - 10 \)

Final Answers

1. \( \boldsymbol{11 + 5c} \)
2. \( \boldsymbol{9n} \)
3. \( \boldsymbol{19 + 2x} \)
4. \( \boldsymbol{5m - 10} \)