Problem 87 - Elementary and Intermediate Algebra

Question

Simplify. $$ 2 z+5(z-3)-10 $$

Step-by-Step Solution

Answer

The simplified expression is \( 7z - 25 \).

Step 1: Distribute the 5

First, distribute the 5 across the terms inside the parentheses. This means applying the distributive property: \[ 5(z - 3) = 5 imes z - 5 imes 3 = 5z - 15 \]Substitute this back into the expression.So, the equation becomes: \[ 2z + 5z - 15 - 10 \]

Step 2: Combine Like Terms

Now, add together all the like terms. Combine the terms with \( z \): \[ 2z + 5z = 7z \]Next, combine the constant terms:\[ -15 - 10 = -25 \]So the expression is simplified to:\[ 7z - 25 \]

Key Concepts

Distributive Property

The **distributive property** is a valuable tool in algebra, used to simplify expressions by distributing a multiplier over terms inside a parenthesis.
If you have an expression like \( a(b + c) \), applying the distributive property means you multiply \( a \) with every term inside the parenthesis:\[a(b + c) = ab + ac\]In the context of our original exercise, we had to distribute the number 5 across the terms \( (z - 3) \). So, we multiplied 5 by \( z \) and by \( -3 \), resulting in two terms:

• 5 times \( z = 5z \)
• 5 times \( -3 = -15 \)

This step is crucial because it breaks down the operation inside the parenthesis and allows us to handle the expression term by term. Distributing effectively sets the stage for further simplification.

Combining Like Terms

After using the distributive property, our next step is to **combine like terms**. Like terms are terms that have the same variable and the same power. This means their coefficients can be added or subtracted.
For example, \( 2z \) and \( 5z \) are like terms because they both have the variable \( z \). When simplifying an expression, these terms can be combined through addition or subtraction:\[2z + 5z = 7z\]Similarly, constants like -15 and -10 are also considered like terms, and they can be combined:

• Additive operation: \(-15 - 10 = -25\)

By combining like terms, the expression becomes more concise, as unnecessary terms are systematically simplified. This streamlines the overall expression and gives a clearer picture of the result.

Algebraic Expressions

An **algebraic expression** is a mathematical phrase that can include numbers, variables, and operators (such as +, -, or ×). In algebra, expressions are simplified to make them easier to understand and work with.
Let's take our example: \(2z + 5(z - 3) - 10\). Initially, it looks complex, but by applying the distributive property and combining like terms, it can be simplified to \(7z - 25\).
Key parts of an algebraic expression include:

• Variables: Letters representing numbers (e.g., \( z\) in our expression).
• Coefficients: The numeric factor of a term involving a variable (e.g., 2 in \( 2z\) and 5 in \( 5z\)).
• Constants: Numeric values without variables (e.g., -10 and -15).

Simplification involves reducing an expression to its most basic form, making it manageable and easier to analyze or solve in equations.