Problem 90 - Beginning and Intermediate Algebra

Question

Use the distributive property to rewrite each expression. $$ -5(2 x-5 y+6 z) $$

Step-by-Step Solution

Answer

-10x + 25y - 30z

Step 1 - Apply the distributive property

Use the distributive property to multiply \(-5\) by each term inside the parentheses.ewline The distributive property states: \[ a(b+c) = ab + ac \]

Step 2 - Multiply \(-5\) by each term

Multiply \(-5\) by \(2x\): \(-5 \cdot 2x = -10x\) ewline Multiply \(-5\) by \(-5y\): \(-5 \cdot -5y = 25y\) ewline Multiply \(-5\) by \(+6z\): \(-5 \cdot 6z = -30z\)

Step 3 - Combine the products

Combine the results of each multiplication to get the final expression: \(-10x + 25y - 30z \)

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operators (like +, -, *, /). These expressions represent mathematical relationships and can be simplified or manipulated using various algebraic rules and properties.
In our exercise, the expression inside the parentheses is an algebraic expression: -5(2x - 5y + 6z).
Here, we have:

• Variables: x, y, z
• Coefficients: 2, -5, 6
• Operators: +, -

Understanding how to work with these expressions is key to solving many algebra problems. Let's explore multiplication and how it plays a role in simplifying these expressions.

Multiplication

Multiplication in algebra involves combining terms and distributing values across an expression. In the context of the distributive property, multiplication helps break down complex expressions into simpler components.
The distributive property states that: For any numbers \(a, b, c\): a(b + c) = ab + ac

Let's see how we apply it in the given exercise:

• Multiply -5 by 2x: -5 \cdot 2x = -10x
• Multiply -5 by -5y: -5 \cdot -5y = 25y
• Multiply -5 by 6z: -5 \cdot 6z = -30z

By distributing -5 to each term inside the parentheses, we transform the original expression into a simpler form. This is a crucial step before we can combine like terms.

Combining Like Terms

Once we have distributed and simplified the terms, the next step is to combine like terms. Like terms are terms with the same variable raised to the same power. This helps us further simplify an expression.
In our exercise, after using the distributive property, we get: -10x + 25y - 30z

Since there are no like terms to combine (each term has a different variable), this is our final simplified expression. If we had like terms, we would add or subtract their coefficients to combine them into one term. Combining like terms streamlines expressions and is especially useful for solving equations and inequalities. Understanding this concept ensures you can simplify any algebraic expression you encounter.