Multiplication Property of Equality
The Multiplication Property of Equality states that multiplying both sides of an equation by the same nonzero quantity preserves the equality — a foundational algebraic principle used in the elimination method in enVision Algebra 1 Chapter 4 for Grade 11. If a = b, then ac = bc for any c ≠ 0. For example, multiplying x + 2y = 8 by 3 gives 3x + 6y = 24. Multiplying 2x - y = 5 by -1 gives -2x + y = -5. This property is the mechanism that creates opposite coefficients before adding equations in the elimination method, making it a core tool for solving systems.
Key Concepts
Multiplication Property of Equality: If both sides of an equation are multiplied by the same nonzero quantity, the solution is unchanged. In symbols, if $a = b$, then $ac = bc$ where $c \neq 0$.
Common Questions
What does the Multiplication Property of Equality state?
If both sides of an equation are multiplied by the same nonzero number, the two sides remain equal. Formally: if a = b, then ac = bc for any c ≠ 0.
What happens when you multiply x + 2y = 8 by 3?
Multiply every term by 3: 3(x) + 3(2y) = 3(8), giving 3x + 6y = 24. The solution set is unchanged.
How is this property used in the elimination method?
To create opposite coefficients, multiply one or both equations by a nonzero constant. For example, to eliminate y from {x + 3y = 7, 2x - y = 4}, multiply the second equation by 3: 6x - 3y = 12. Now the y terms are +3y and -3y, which add to zero.
Why must the multiplier c be nonzero?
Multiplying both sides by 0 gives 0 = 0, which loses all information about the variable. The equation becomes trivially true but reveals nothing about the solution.
Can you divide both sides by the same nonzero number?
Yes. Division by c is the same as multiplication by 1/c, so the Multiplication Property of Equality covers division as well (as long as you divide by a nonzero number).