Problem 72 - Beginning and Intermediate Algebra

Question

Simplify each expression. \(5 x+3(7-2 x)\)

Step-by-Step Solution

Answer

-x + 21

Step 1 - Distribute the Constant

First, distribute the constant 3 inside the parenthesis. The expression inside the parenthesis is (7 - 2x). Multiply 3 by each term inside: 3 * 7 and 3 * (-2x)

Step 2 - Perform the Multiplication

Now perform the multiplication: 3 * 7 = 21 and 3 * (-2x) = -6x. This transforms the original expression into: 5x + 21 - 6x.

Step 3 - Combine Like Terms

Combine the like terms, which are 5x and -6x. Use the rule of addition: 5x - 6x = -1x or simply -x. So, the resulting expression is: -x + 21.

Key Concepts

Distributive Property

The distributive property is a foundational concept in algebra. It states that multiplying a number by a sum is the same as doing each multiplication separately. For example, in the expression 3(7 - 2x), we apply the distributive property.To break it down:

• First, multiply the constant 3 by each term inside the parentheses.
• Perform the calculations separately: 3 * 7 = 21 and 3 * (-2x) = -6x.

Now, the expression of 3(7 - 2x) becomes 21 - 6x. This step prepares the expression for further simplification.

Combining Like Terms

Combining like terms is another essential skill in algebra. Like terms are terms that have the same variable raised to the same power. For instance, 5x and -6x are like terms because they both contain the variable x.When we have an expression like 5x + 21 - 6x, we can combine 5x and -6x. Here’s how:

• Add or subtract the coefficients (the numbers in front of the variables).
• In this case, 5x - 6x = -1x, which simplifies to just -x.

After combining the like terms, the expression 5x + 21 - 6x becomes -x + 21. Simplifying in this way makes the expression easier to interpret and solve.

Order of Operations

The order of operations is a rule to solve any mathematical expression. It ensures consistent and correct results when simplifying expressions. The typical order can be remembered with the acronym PEMDAS:

• Parentheses first
• Exponents (i.e., powers and roots etc.)
• Multiplication and Division (left to right)
• Addition and Subtraction (left to right)

In the exercise, we applied PEMDAS by starting inside the parentheses: (7 - 2x). Then, we distributed the 3 through multiplication. Afterward, we combined like terms as the final step in simplification rules. Following order of operations assures us that we arrive at the correct and simplified form of the expression with proper mathematical conventions.