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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

1 answer

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: \[ x = -6 \]