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8.2 Solve Equations Using the Division and Multiplication Properties of Equality

3 min readLast Updated on June 25, 2024

Solving equations is all about balance. By using multiplication and division properties of equality, we can keep both sides of an equation equal while isolating variables. This helps us find unknown values and solve problems.

These properties are the foundation for solving more complex equations. By combining like terms and applying these properties step-by-step, we can tackle multi-step equations and simplify expressions to find solutions.

Solving Equations Using Division and Multiplication Properties of Equality

Equation solving with division or multiplication

Top images from around the web for Equation solving with division or multiplication
Top images from around the web for Equation solving with division or multiplication
  • Multiplication Property of Equality maintains the truth of an equation when both sides are multiplied or divided by the same non-zero number
    • Isolates a variable by multiplying or dividing both sides by the reciprocal of the variable's coefficient (3x=123x = 12, multiply both sides by 13\frac{1}{3} to get x=4x = 4)
  • Division Property of Equality maintains the equality when dividing both sides of an equation by the same non-zero number
    • Solves for a variable by dividing both sides by the variable's coefficient (x4=5\frac{x}{4} = 5, multiply both sides by 4 to get x=20x = 20)
  • Isolating the variable gets the variable alone on one side of the equation
    • Uses the Multiplication or Division Property of Equality to eliminate the variable's coefficient (2x+3=112x + 3 = 11, subtract 3 from both sides, then divide both sides by 2 to get x=4x = 4)
  • These properties are fundamental in solving algebraic equations

Simplifying equations with like terms

  • Like terms have the same variables raised to the same powers and can be added or subtracted to simplify an equation (3x+2x=5x3x + 2x = 5x)
  • Combining like terms simplifies the equation by adding or subtracting the coefficients of like terms on both sides separately (2x+3+x=4x12x + 3 + x = 4x - 1, simplify to 3x+3=4x13x + 3 = 4x - 1)
  • Solving simplified equations uses the properties of equality to solve for the variable after combining like terms (3x+3=4x13x + 3 = 4x - 1, subtract 3x3x from both sides, then subtract 3 from both sides to get x=2x = 2)

Properties of equality for multi-step equations

  • Properties of equality:
    1. Reflexive Property: a=aa = a
    2. Symmetric Property: If a=ba = b, then b=ab = a
    3. Transitive Property: If a=ba = b and b=cb = c, then a=ca = c
    4. Addition Property: If a=ba = b, then a+c=b+ca + c = b + c
    5. Subtraction Property: If a=ba = b, then ac=bca - c = b - c
  • Solving multi-step equations:
    1. Simplify each side of the equation by combining like terms
    2. Use the Addition or Subtraction Property of Equality to isolate the variable term
    3. Use the Multiplication or Division Property of Equality to eliminate the variable's coefficient (2(3x1)+4=5x+72(3x - 1) + 4 = 5x + 7, simplify to 6x2+4=5x+76x - 2 + 4 = 5x + 7, then 6x+2=5x+76x + 2 = 5x + 7, subtract 5x5x from both sides to get x+2=7x + 2 = 7, then subtract 2 from both sides to get x=5x = 5)

Additional Concepts

  • Order of operations is crucial when simplifying equations and expressions
  • The distributive property helps expand expressions: a(b+c)=ab+aca(b + c) = ab + ac
  • Reciprocals are used in division, where multiplying by the reciprocal is equivalent to dividing: ab×ba=1\frac{a}{b} \times \frac{b}{a} = 1

Term 1 of 15

Algebraic Equations
See definition

Algebraic equations are mathematical statements that express the relationship between variables and constants using algebraic operations. They are fundamental to solving problems in pre-algebra and algebra, as they allow for the manipulation and simplification of expressions to find unknown values.

Key Terms to Review (15)

Term 1 of 15

Algebraic Equations
See definition

Algebraic equations are mathematical statements that express the relationship between variables and constants using algebraic operations. They are fundamental to solving problems in pre-algebra and algebra, as they allow for the manipulation and simplification of expressions to find unknown values.

© 2025 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.

Term 1 of 15

Algebraic Equations
See definition

Algebraic equations are mathematical statements that express the relationship between variables and constants using algebraic operations. They are fundamental to solving problems in pre-algebra and algebra, as they allow for the manipulation and simplification of expressions to find unknown values.



© 2025 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.

© 2025 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.