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Chapter 1: Problem 99

Simplify. $$ 15 x-y-5(3 x-2 y+5 z) $$

Short Answer

Expert verified
9y - 25z

Step by step solution

01

Distribute the -5

First, distribute the \textbf{-5} to each term inside the parentheses. This means you will multiply \textbf{-5} by \textbf{3x}, \textbf{-2y}, and \textbf{5z}:\[ \text{-5(3x) = -15x} \] \[ \text{-5(-2y) = 10y} \] \[ \text{-5(5z) = -25z} \]
02

Rewrite the expression

Now we can rewrite the expression by combining all the results from Step 1. Replace the original expression with the new terms:\[ 15x - y - 15x + 10y - 25z \]
03

Combine like terms

Combine like terms from the new expression:\[ \text{The } 15x \text{ and } -15x \text{ cancel each other out} \] \[ - y + 10y = 9y \] \[ \text{The } -25z \text{ remains the same} \] So the simplified expression is:\[ 9y - 25z \]

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Key Concepts

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

Algebraic Expressions
Algebraic expressions are like sentences in the language of algebra. They are made up of numbers, variables (like x or y), and arithmetic operations (like addition or multiplication).

For example, in the expression $$15x - y - 5(3x - 2y + 5z)$$:
  • \textbf{15x}, \textbf{y}, and \textbf{z} are terms.
  • The numbers 15, -5, 3, -2, and 5 are coefficients.
  • The letters x, y, and z are variables, representing unknown values.
When working with algebraic expressions, we perform operations like addition, subtraction, and distribution to simplify or manipulate them.
Distributive Property
The distributive property helps simplify expressions with parentheses. It states that for any numbers or variables, \textbf{a(b + c) = ab + ac}. This means we multiply each term inside the parentheses by the number outside.

In the given exercise, we apply the distributive property to the expression $$-5(3x - 2y + 5z)$$:
  • \textbf{-5 times 3x} gives us \textbf{-15x}.
  • \textbf{-5 times -2y} gives us \textbf{10y} (since two negatives make a positive).
  • \textbf{-5 times 5z} gives us \textbf{-25z}.
So, the expression inside the parentheses is transformed into \textbf{-15x + 10y - 25z}.
Simplifying Expressions
To simplify an expression, we combine like terms. Like terms are terms that have the same variable raised to the same power.

In the exercise, after distributing, we have the expression $$15x - y - 15x + 10y - 25z$$. We simplify this by:
  • Combining the \textbf{15x} and \textbf{-15x}, which cancel each other out.
  • Combining the \textbf{-y} and \textbf{10y} to get \textbf{9y}.
  • The \textbf{-25z} remains unchanged.
Thus, the simplified expression is \textbf{9y - 25z}.

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