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Chapter 2: Problem 80

Simplify each expression. \(2(4 k-7)-4(-k+3)\)

Short Answer

Expert verified
Simplified expression is \[12k - 26\].

Step by step solution

01

- Distribute the Constants

Distribute the constants outside the parentheses to each term inside the parentheses: First, distribute the 2 in the expression: \[2(4k - 7) = 2 \times 4k + 2 \times (-7) = 8k - 14\] Next, distribute the -4 in the expression: \[-4(-k + 3) = -4 \times (-k) + (-4) \times 3 = 4k - 12\]
02

- Combine the Expressions

Now, combine the expressions from Step 1: \[8k - 14 + 4k - 12\]
03

- Combine Like Terms

Combine the like terms (terms with the same variable): Combine the 'k' terms: \[8k + 4k = 12k\] Combine the constant terms: \[-14 - 12 = -26\] This gives us the simplified expression: \[12k - 26\]

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Key Concepts

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

Distributive Property
The distributive property is a fundamental concept in algebra that involves multiplying a single term by each term inside a set of parentheses. This property helps to simplify expressions and solve equations. In the given exercise, we apply the distributive property to the expression: \(2(4k - 7) - 4(-k + 3)\). Here鈥檚 how it works:

  • First, distribute the 2 across the terms within the first parentheses: \(2(4k - 7) = 8k - 14\).
  • Next, distribute -4 across the terms within the second parentheses: \(-4(-k + 3) = 4k - 12\).
By using the distributive property, we've transformed a complex expression into simpler parts. This is the first step in simplifying any algebraic expression.
Combining Like Terms
Combining like terms is an essential skill in algebra. It involves merging terms that have the same variable and exponent. This process helps to further simplify expressions. After distributing terms in the given exercise, we have the expression: \(8k - 14 + 4k - 12\). The next step is to combine like terms:

  • Combine the 'k' terms: \(8k + 4k\) which gives us \(12k\).
  • Combine the constant terms: \(-14 - 12\) which gives us \(-26\).
By combining these like terms, we reduce the expression to \(12k - 26\). This reduction simplifies the equation, making it easier to understand and solve.
Linear Equations
Linear equations are equations of the first degree, meaning each term is either a constant or the product of a constant and a single variable. In our simplified expression \(12k - 26\), we see an example of a linear equation.

Some important properties of linear equations include:
  • They graph as straight lines.
  • They have variables raised only to the first power.
  • They can be solved by isolating the variable on one side of the equation.
Understanding linear equations is crucial for solving algebra problems, as they form the basis for more complex mathematical concepts. Our final expression, \(12k - 26\), exemplifies a linear equation in its simplified form, demonstrating how to transform and solve such expressions effectively.

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