Question

14+2(4x+5y)−x−2(y−6)=x−2(5−4y) Identify the following parts of the equation: the expression(s), variable(s), coefficient(s), constant(s), factor(s) and terms. SOLVE the equation. While solving, identify if and when you use the following properties: Commutative Properties, Associative Properties, Distributive Property

Answer 1

To analyze the given equation and solve it, let's clearly define the parts as requested.

### Given Equation:
\[ 14 + 2(4x + 5y) - x - 2(y - 6) = x - 2(5 - 4y) \]

### Identify Parts of the Equation
1. **Expressions**:
- Left Side: \( 14 + 2(4x + 5y) - x - 2(y - 6) \)
- Right Side: \( x - 2(5 - 4y) \)

2. **Variables**:
- \( x \)
- \( y \)

3. **Coefficients**:
- For \( x \): \( -1 \) (from \(-x\)) and \( 1 \) (from \(x\) on the right side)
- For \( y \): \( 2 \) (from \(10y\) on the left side after distribution)

4. **Constants**:
- \( 14 \) (from the left side)
- \( -12 \) (result of distributing \(-2\) and combining constants)

5. **Factors** (related to multiplication):
- \( 2 \) (as a factor in \(2(4x + 5y)\))
- \( 4 \), \( 5 \) (as factors in \(2(4x + 5y)\) and \(-2(5 - 4y)\))
- \( -2 \) (as a factor in \(-2(y - 6)\) and \(-2(5 - 4y)\))

6. **Terms**:
- \( 14 \)
- \( 2(4x + 5y) \)
- \( -x \)
- \( -2(y - 6) \)
- \( x \)
- \( -2(5 - 4y) \)

### Solve the Equation

1. **Apply the Distributive Property**:
\[
14 + 2(4x + 5y) - x - 2(y - 6) = x - 2(5 - 4y)
\]
Distribute \(2\) and \(-2\):
\[
14 + 8x + 10y - x - 2y + 12 = x - 10 + 8y
\]

2. **Combine Like Terms** on the left side:
\[
(8x - x) + (10y - 2y) + (14 + 12) = x - 10 + 8y
\]
Simplifying:
\[
7x + 8y + 26 = x - 10 + 8y
\]

3. **Eliminate \(8y\)** from both sides (using the Commutative Property here):
\[
7x + 26 = x - 10
\]

4. **Move the \(x\)** from the right to the left to consolidate \(x\) terms:
\[
7x - x + 26 = -10
\]
This simplifies to:
\[
6x + 26 = -10
\]

5. **Subtract 26** from both sides:
\[
6x = -10 - 26
\]
\[
6x = -36
\]

6. **Divide by 6**:
\[
x = -6
\]

### Summary of Steps and Properties Used
- **Distributive Property** was used when expanding \(2(4x + 5y)\) and \(-2(y - 6)\).
- **Commutative Property** was used for rearranging terms when canceling \(8y\).
- **Associative Property** was not directly used, but is inherent in the grouping of terms.

### Final Solution
The solution to the equation is:
\[ x = -6 \]