Equations with Parentheses: Solving Using Distributive Property

Overview

• Focus on solving equations with one variable using the distributive property.
• Steps involve distributing, combining like terms, and isolating variables.

Example 1

Equation

• Original equation: 10p - 3 = 2(12 + 4p) - 7

Steps

1. Distribute

   • Distribute 2 over (12 + 4p):
     • 2 * 12 = 24
     • 2 * 4p = 8p
   • Resulting equation: 10p - 3 = 24 + 8p - 7

2. Combine Like Terms

   • On the right: 24 - 7 = 17
   • New equation: 10p - 3 = 17 + 8p

3. Isolate Variable

   • Subtract 8p from both sides: 10p - 8p = 2p
   • New equation: 2p - 3 = 17
   • Add 3 to both sides: 2p = 20
   • Divide by 2: p = 10

Example 2

Equation

• Original equation: 18 - 3f - 4 = 3f - 4

Steps

1. Distribute

   • Distribute 3 over (6 - f):
     • 3 * 6 = 18
     • 3 * (-f) = -3f

2. Combine Like Terms

   • Simplify: 18 - 3f - 4 = 14 - 3f
   • Equation becomes: 14 - 3f = 3f - 4

3. Isolate Variable

   • Add 3f to both sides: 14 = 6f - 4
   • Add 4 to both sides: 18 = 6f
   • Divide by 6: f = 3

Example 3

Equation

• Simplified equation: -t = 9(t - 10)

Steps

1. Distribute

   • Distribute 9 over (t - 10):
     • 9 * t = 9t
     • 9 * (-10) = -90
   • Equation: -t = 9t - 90

2. Isolate Variable

   • Subtract 9t from both sides: -t - 9t = -90
   • Simplify: -10t = -90
   • Divide by -10: t = 9

Summary

• Key Steps:
  • Distribute coefficients across terms in parentheses.
  • Combine like terms to simplify equation.
  • Isolate variable by moving terms across the equation using addition/subtraction.
  • Solve for the variable by dividing the coefficients.