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Chapter 3: Problem 2

Define the key terms. a. Division property of equality b. Multiplication property of equality

Short Answer

Expert verified
Division: If a = b, then \frac{a}{c} = \frac{b}{c} for c \eq 0. Multiplication: If a = b, then ac = bc.

Step by step solution

01

- Define division property of equality

The division property of equality states that if both sides of an equation are divided by the same nonzero number, the two sides remain equal. Mathematically, if \(a = b \) then \[ \frac{a}{c} = \frac{b}{c} \ \text{, for } c \eq 0 \]
02

- Define multiplication property of equality

The multiplication property of equality states that if both sides of an equation are multiplied by the same number, the two sides remain equal. Mathematically, if \(a = b \) then \[ ac = bc \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

division property of equality
The division property of equality is a fundamental concept in algebra. It states that if you have an equation where two expressions are equal, you can divide both sides by the same nonzero number and the equation will still hold true. This is crucial because it helps us to isolate the variable and solve for it. For example, if you have the equation 10 = 10 and you divide both sides by 2, you get 5 = 5, which is still a true statement. Remember, the number you're dividing by must not be zero, because division by zero is undefined. Always double-check your operations to make sure this rule is followed.
multiplication property of equality
The multiplication property of equality is another basic yet important principle in algebra. This property states that if you have an equation and you multiply both sides by the same number, the equation remains true. For example, if you start with an equation like 3 = 3 and multiply both sides by 4, you get 12 = 12, which is still balanced. This property is extremely useful for solving equations, especially when you are trying to clear fractions or decimals. Just be careful to multiply all terms on both sides of the equation, and you will maintain equality.
equations
An equation is a mathematical statement that asserts the equality of two expressions. In algebra, equations often include variables鈥攕ymbols that represent unknown values鈥攖hat you need to solve for. A simple example of an equation is 2x + 3 = 7. To solve such equations, you use various properties of equality, such as the division and multiplication properties, to isolate the variable and find its value. Equations can be equalities, inequalities, linear, quadratic, and much more. They are foundational to much of algebra and higher mathematics, so understanding how to manipulate and solve them is key to success in these subjects.

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Most popular questions from this chapter

Clear parentheses and combine like terms. $$4+6(y-3)$$

Solve the equation. $$-5+7=5 x-4(x-1)$$

Apply the distributive property. $$-2(p+4)$$

Clear parentheses and combine like terms. $$6(y-9)-5(2 y-5)$$

Clear parentheses and combine like terms. $$10(x+5)-3(2 x+9)$$
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