Question

The distributive property.
1. 7-4 [3 - (4y - 5)]
2. 6-5[8 - (2y - 4)]
3. 5x - 3 [7 - 2 (6x -7) -3x)]
4. 6+3 [2x - 4(3x - 2)]

Answer 1

To simplify the expressions using the distributive property, we will expand them step by step. The distributive property states that a(b + c) = ab + ac.

### 1. \( 7 - 4[3 - (4y - 5)] \)

First, simplify the expression inside the parentheses:

\[
4y - 5 \rightarrow -(4y - 5) = -4y + 5
\]

So:

\[
3 - (4y - 5) = 3 - 4y + 5 = 8 - 4y
\]

Now, substitute back into the expression:

\[
7 - 4[8 - 4y]
\]

Now distribute \(-4\):

\[
7 - 4 \cdot 8 + 4 \cdot 4y = 7 - 32 + 16y
\]

Combine the constant terms:

\[
7 - 32 = -25
\]

Final result:

\[
-25 + 16y
\]

### 2. \( 6 - 5[8 - (2y - 4)] \)

First, simplify inside the parentheses:

\[
2y - 4 \rightarrow -(2y - 4) = -2y + 4
\]

So:

\[
8 - (2y - 4) = 8 + 2y - 4 = 4 + 2y
\]

Now substitute back into the expression:

\[
6 - 5[4 + 2y]
\]

Now distribute \(-5\):

\[
6 - 5 \cdot 4 - 5 \cdot 2y = 6 - 20 - 10y
\]

Combine the constant terms:

\[
6 - 20 = -14
\]

Final result:

\[
-14 - 10y
\]

### 3. \( 5x - 3[7 - 2(6x - 7) - 3x] \)

First, simplify the expression inside the parentheses:

\[
2(6x - 7) = 12x - 14
\]

Now substitute into the expression:

\[
5x - 3[7 - (12x - 14) - 3x]
\]

Now simplify inside the brackets:

\[
7 - 12x + 14 - 3x = 21 - 15x
\]

Now substitute back:

\[
5x - 3[21 - 15x]
\]

Distribute \(-3\):

\[
5x - 63 + 45x
\]

Combine like terms:

\[
5x + 45x - 63 = 50x - 63
\]

Final result:

\[
50x - 63
\]

### 4. \( 6 + 3[2x - 4(3x - 2)] \)

First, simplify the expression inside the brackets:

\[
4(3x - 2) = 12x - 8
\]

Now substitute back:

\[
6 + 3[2x - (12x - 8)]
\]

Now simplify inside the brackets:

\[
2x - 12x + 8 = -10x + 8
\]

Now substitute back:

\[
6 + 3[-10x + 8]
\]

Distribute \(3\):

\[
6 - 30x + 24
\]

Combine the constants:

\[
30 - 30x
\]

Final result:

\[
30 - 30x
\]

### Summary of Results:
1. \( -25 + 16y \)
2. \( -14 - 10y \)
3. \( 50x - 63 \)
4. \( 30 - 30x \)