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Chapter 15: Problem 61

Multiply. $$ 4(2 x+7) $$

Short Answer

Expert verified
8x + 28

Step by step solution

01

Understand the Expression

The given expression is 4(2x + 7). Here, the number 4 is being multiplied by the entire expression inside the parentheses.
02

Apply the Distributive Property

Use the distributive property of multiplication to distribute the 4 to each term inside the parentheses. This means multiplying 4 by 2x and then by 7.
03

Multiply the Terms Individually

First, multiply 4 by 2x, which gives 8x. Next, multiply 4 by 7, which gives 28.
04

Combine the Results

Combine the results of the individual multiplications. Therefore, 4(2x + 7) = 8x + 28.

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Key Concepts

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

Distributive Property in Multiplication
The Distributive Property is a fundamental concept in algebra. It allows us to multiply a single term by every term within a set of parentheses. In the exercise, we have the expression 4(2x + 7). Using the distributive property, we distribute the 4 to both 2x and 7.

This means:
  • Multiply 4 by 2x (4 * 2x )

  • Then multiply 4 by 7 (4 * 7 )

Therefore, 4(2x + 7) becomes 8x + 28. This step-by-step approach helps simplify complex algebraic expressions and makes solving them manageable.
Understanding Algebraic Expressions
Algebraic expressions are mathematical phrases that can include numbers, variables, and arithmetic operations. In our exercise, the expression inside the parentheses is 2x + 7.

Here,
  • '2x' is a term where 2 is the coefficient, and 'x' is the variable.

  • '7' is a constant term.

Understanding these elements allows us to manipulate and simplify expressions correctly. By applying the Distributive Property, we handle each term logically, ensuring every component is accounted for in the solution.
Combining Like Terms
Combining like terms is a technique used to simplify algebraic expressions. Like terms are terms that have the same variable raised to the same power.

In the final step of our solution, we combine the results of the distributed products:
  • 8x (which comes from 4 * 2x)

  • 28 (which comes from 4 * 7)

Since 8x and 28 are unlike terms (one has a variable, the other doesn't), they cannot be combined further. Thus, the simplified expression is 8x + 28. Recognizing and combining like terms is critical for correct and simplified final answers in algebra.

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Most popular questions from this chapter

E-Chem Testing has a solution that is \(80 \%\) base and another that is \(30 \%\) base. A technician needs \(200 \mathrm{L}\) of a solution that is \(62 \%\) base. The \(200 \mathrm{L}\) will be prepared by mixing the two solutions on hand. How much of each should be used?

Solve each system using the elimination method. $$ \begin{aligned} x-2 y &=6 \\ -x+3 y &=-4 \end{aligned} $$

Remove parentheses and simplify. $$ 2[2 a+1-(3 a-6)] $$

Multiply. $$ -7(3 x-5) $$

Solve each system graphically. $$ \begin{aligned} &x+y=7\\\ &x-y=1 \end{aligned} $$
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