Problem 42 - Beginning and Intermediate Algebra

Question

Use the distributive property to write each expression without parentheses. Then simplify the result. See Example 4. \(-3(z-y)\)

Step-by-Step Solution

Answer

The expression simplifies to \(-3z + 3y\).

Step 1: Identify the Expression

The expression given is \(-3(z-y)\). To simplify, we need to use the distributive property to eliminate the parentheses.

Step 2: Apply the Distributive Property

The distributive property states that \(a(b-c) = ab - ac\). Therefore, distribute \(-3\) to both \(z\) and \(-y\). This gives us: \[-3(z-y) = -3z + 3y.\]

Step 3: Simplify the Expression

Since \(-3z + 3y\) is already simplified to its simplest form, nothing further is required. The expression is simplified as \(-3z + 3y\).

Key Concepts

Simplifying Algebraic Expressions

Simplifying algebraic expressions involves making them as concise and uncomplicated as possible. When we have an expression like \(-3(z-y)\), we first need to get rid of any parentheses. This is often done by applying properties like the distributive property, which helps "distribute" or "spread out" a number across terms inside the parentheses. After applying the distributive property, we get an expression made up of simpler terms, like in our example: \(-3z + 3y\), without parentheses and already in its simplest form. It involves combining like terms and ensuring that the expression is as reduced as possible. Simplifying expressions helps in making calculations easier and in finding the value of the expression for different variable values. Some tips for simplifying are:

• Always perform operations according to the order of operations (use PEMDAS/BODMAS as a guide).
• Look for like terms that can be combined.
• Use algebraic properties to rearrange and simplify terms.

Doing this reduces errors and makes solving algebraic problems much smoother.

Negative Numbers

Understanding negative numbers is vital when simplifying algebraic expressions, especially in algebra. In our exercise, \(-3(z-y)\), the negative sign in front of the 3 affects both terms inside the parentheses, z and -y.Negative numbers mean the opposite or inverse of a number on the number line. When distributing a negative number, it "flips" the sign of whatever term it's multiplied by. For example:

• Distributing -3 to z results in -3z.
• Distributing -3 to -y results in +3y (since negative times negative is positive).

This is why it is crucial in algebra to mind the signs for each number or term since they can completely change the expression's value.Using negative numbers correctly ensures the expression stays accurate and reflects the intended calculation. Often, mistakes happen by overlooking the change in signs, so it's important to be cautious and thoughtful about these when simplifying.

Algebraic Properties

Algebraic properties are rules that govern the operations of algebra and make it easier to simplify and solve expressions or equations. For instance, the distributive property is a key tool frequently used to simplify expressions like \(-3(z-y)\). The distributive property is expressed as \(a(b-c) = ab - ac\). It allows you to "distribute" a factor across terms inside parentheses, helping eliminate parentheses and simplify expressions. But that's not the only property; there are others like:

• Associative Property: which deals with grouping, like how \((a + b) + c = a + (b + c)\).
• Commutative Property: which deals with the order, like \(a + b = b + a\).

These properties are the foundational rules for manipulating and simplifying algebraic expressions. They help reorder, reorganize, and solve, making it clearer to perform operations correctly. Learning these properties allows you to simplify and solve equations and expressions faster and with more accuracy. By applying them wisely, we ensure our work is mathematically sound and less prone to error.