If $x=y$, then $x z=y z$ represents the ______ property of equality.
A. addition
B. division
C. multiplication
D. subtraction
Answer (1)
The problem involves identifying the property of equality that transforms x = y into x z = yz .
The addition property of equality states that if x = y , then x + z = y + z .
The subtraction property of equality states that if x = y , then x − z = y − z .
The multiplication property of equality states that if x = y , then x z = yz , which matches the given transformation. Therefore, the answer is multiplication .
Explanation
Understanding the Problem We are given the equation x = y and we want to determine which property of equality is demonstrated by the transformation to x z = yz . The properties of equality include addition, subtraction, multiplication, and division. Let's examine each one.
Addition Property The addition property of equality states that if x = y , then x + z = y + z . This is not the transformation we see.
Subtraction Property The subtraction property of equality states that if x = y , then x − z = y − z . This is also not the transformation we see.
Multiplication Property The multiplication property of equality states that if x = y , then x z = yz . This exactly matches the transformation we are given.
Division Property The division property of equality states that if x = y , then z x = z y , provided z = 0 . This is not the transformation we see.
Conclusion Since the transformation x = y to x z = yz matches the multiplication property of equality, the answer is multiplication.
Examples
The multiplication property of equality is a fundamental concept in algebra and is used in many real-world applications. For example, if you know that two quantities are equal, and you need to scale them by the same factor, you can use the multiplication property of equality to ensure that the scaled quantities remain equal. Imagine you're baking a cake and the recipe calls for twice as many ingredients. If you double each ingredient, you're using the multiplication property of equality to maintain the correct proportions and ensure the cake turns out as expected. This principle applies in various scenarios, from scaling recipes to adjusting financial budgets proportionally.
