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Chapter 1: Problem 15

$$\text { Simplify the given algebraic expressions.}$$ $$v-(4-5 x+2 v)$$

Short Answer

Expert verified
The simplified expression is \(-v + 5x - 4\).

Step by step solution

01

Distribute Negative Sign

The expression is: \( v-(4-5x+2v) \). The first step is to distribute the minus sign across the terms inside the parenthesis. This changes each sign: \( v - 4 + 5x - 2v \).
02

Combine Like Terms

Now that the expression is out of its parentheses, combine like terms. We have \( v \) and \(-2v\) to combine: \( v - 2v = -v \). The expression is now \(-v + 5x - 4\).

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Key Concepts

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

Simplifying Expressions
Simplifying algebraic expressions is a fundamental technique in algebra. It involves making expressions easier to work with by reducing them to their simplest form. This means that we aim to eliminate any unnecessary terms or operations. Here鈥檚 how simplification typically works:
  • First, remove any parentheses through distribution or by applying rules such as the order of operations.
  • Then, combine like terms which are terms that have identical variable parts.
  • Finally, reorder the terms for clarity if needed, typically placing higher powers or variables first.

Simplifying makes solving equations manageable and provides clearer interpretation of mathematical models. Always remember to perform operations carefully to avoid errors, ensuring that each step is verified for accuracy.
Negative Sign Distribution
Distributing a negative sign, or minus sign, is an essential skill in algebra when working with expressions that include parentheses. When a negative sign is in front of a set of parentheses, it affects the entire expression inside. To correctly distribute the negative sign, you need to:
  • Change the sign of every term inside the parentheses. Essentially, you are multiplying each term by (-1).
  • For example, consider the expression \( v - (4 - 5x + 2v) \). Distributing the negative gives:\( v - 4 + 5x - 2v \).
  • By doing so, the overall expression outside of the parentheses becomes straightforward to simplify further.

This process can help to remove parentheses altogether, making the expression simpler and paving the way for combining like terms.
Combining Like Terms
Combining like terms is the process of simplifying expressions by merging terms that have identical variable components. When terms look alike, and by alike, we mean they have the exact same variables raised to the same powers, they can be combined by adding or subtracting their coefficients.
  • For example, in the expression \( v - 4 + 5x - 2v \), we can combine the terms \( v \) and \(-2v \) because they each contain the same variable \( v \) raised to the power of 1.
  • This combination gives us \( -v \), since \( 1v + (-2v) = -v \).
  • The updated expression becomes \(-v + 5x - 4 \), with the other terms left as they initially appear because they do not have like terms to combine with.

By applying this rule, you simplify the expression further, which is essential for solving or evaluating algebraic problems efficiently.

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Most popular questions from this chapter

Solve the given problems. A TV screen is 52.3 in. wide and 29.3 in. high. The length of a diagonal (the dimension used to describe it - from one corner to the opposite corner) is found by evaluating \(\sqrt{w^{2}+h^{2}}\), where \(w\) is the width and \(h\) is the height. Find the diagonal.

Perform the indicated multiplications. By multiplication, show that \((x+y)\left(x^{2}-x y+y^{2}\right)=x^{3}+y^{3}\).

Solve the given problems. Refer to Appendix B for units of measurement and their symbols. Describe the values of \(x\) and \(y\) for which (a) \(-x y=1\) and (b) \(\frac{x-y}{x-y}=1\).

Solve the given problems. Refer to Appendix B for units of measurement and their symbols. The daily high temperatures (in ^ \(\mathrm{C}\) ) for Edmonton, Alberta, in the first week in March were recorded as \(-7,-3,2,3,1,-4,\) and -6 What was the average daily temperature for the week? (Divide the algebraic sum of readings by the number of readings.)

Perform the indicated multiplications. Simplify the expression \(\left(T^{2}-100\right)(T-10)(T+10),\) which arises when analyzing the energy radiation from an object.
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