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Chapter 1: Problem 89

Multiply: \(7(3 z+y+2)\).

Short Answer

Expert verified
The result is \(21z + 7y + 14\).

Step by step solution

01

- Distribute the 7

Multiply 7 by each term inside the parentheses. These terms are 3z, y, and 2.
02

- Multiply 7 by 3z

Perform the multiplication: \(7 \times 3z = 21z\)
03

- Multiply 7 by y

Perform the multiplication: \(7 \times y = 7y\)
04

- Multiply 7 by 2

Perform the multiplication: \(7 \times 2 = 14\)
05

- Combine the Products

Combine all the products obtained: \(21z + 7y + 14\)

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Key Concepts

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

Multiplication of Terms
In algebra, multiplying terms is a basic but crucial operation. It's essential to distribute a multiplier, like the number 7 in our exercise, to each term inside the parentheses. This concept is known as the distributive property. For the problem, you start by multiplying 7 with each individual term inside the parentheses: 3z, y, and 2. This operation ensures each term is scaled by the same factor.
For example:
- Multiply 7 by 3z: \( 7 \times 3z = 21z \)
- Multiply 7 by y: \( 7 \times y = 7y \)
- Multiply 7 by 2: \( 7 \times 2 = 14 \)
Understanding and mastering this concept makes solving more complex algebraic expressions easier.
Algebraic Expressions
An algebraic expression is a combination of numbers, variables, and operational symbols (like + or -). In the problem, \(3z + y + 2\) is an algebraic expression. When you multiply this expression by 7, each term within the expression is individually multiplied by 7.
Here are the terms in the expression:
- 3z, where 3 is the coefficient and z is the variable
- y, which is a variable without a coefficient (implying it's 1y)
- 2, which is a constant term
This results in a new algebraic expression: \(21z + 7y + 14\). Being able to identify and manipulate algebraic expressions is a fundamental skill in algebra.
Combining Like Terms
In the final step, we look at combining like terms. However, in our example \(21z + 7y + 14\), there are no like terms to combine. Nonetheless, it's crucial to understand what 'like terms' means in algebra.
Like terms are terms that have the same variable raised to the same power. For instance, \(3x\) and \(5x\) can be combined because they both have the variable x. But \(3x\) and \(3x^2\) can't be combined because the exponents are different.
Whenever you have an expression, always look for terms that can be simplified by combining them. It helps make the expression simpler and more manageable.

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