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Chapter 3: Problem 43

Simplify the expression. $$-(-3 y-4+4 x)$$

Short Answer

Expert verified
The simplified expression is \(3y - 4x + 4\).

Step by step solution

01

Remove the Negative Sign Outside the Parenthesis

To simplify the expression, start by distributing the negative sign (which is equivalent to multiplying by -1) into the terms inside the parenthesis. \(-(-3y-4+4x)\) becomes \(3y + 4 - 4x\).
02

Rearrange the Expression

Since the expression does not require further simplification, you can arrange the terms if you desire a standard form. A typical linear expression arranges variables in alphabetical order:\(3y - 4x + 4\)

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Key Concepts

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

Distributive Property
The concept of the distributive property is fundamental in algebra and makes simplifying expressions much easier. Let's break it down. The distributive property states that any number multiplied by a sum is equal to the sum of that number multiplied by each addend separately. In mathematical terms, this means:
  • For any numbers or expressions, if you have something like \(a(b + c)\), it will simplify to \(ab + ac\).
In the given exercise, \(-(-3y-4+4x)\), the negative sign outside the parenthesis can be interpreted as multiplying by -1. Therefore, you apply the distributive property:
  • Multiply each term inside the parenthesis by -1
  • This changes each sign: \(3y\) becomes \(+3y\)\, \(-4\) turns into \(+4\)\, and \(4x\) becomes \(-4x\).
Working through this step, the expression transforms to \(3y + 4 - 4x\), making it simpler to manage.
Linear Expressions
Linear expressions are a type of algebraic expression where each term is either a constant or the product of a constant and a single variable. They are called "linear" because their graph represents a straight line. Let's see what that means:
  • The standard form of a linear expression in the context of algebra is \(ax + by + c\), where \(a\), \(b\), and \(c\) are constants, and \(x\) and \(y\) are variables.
In the expression from the exercise, \(3y - 4x + 4\), we see it meets this criterion as each term is either a number (constant) or a variable part. Linear expressions are straightforward to handle and are the building blocks for more complex algebraic expressions. They form the basis of equations and inequalities that model real-world situations like calculating distances, budgeting, and so on.
Rearranging Terms
Rearranging terms in an expression is often a helpful strategy to organize or simplify your work further. By putting variables and constants in a common or more standardized order, it becomes easier to read and understand. Here's how it works:
  • Begin by identifying similar types of elements:
    • Constants are standalone numbers.
    • Variables are influenced by their coefficients.
  • Shifting pieces around is mainly about making it more conventional or visually agreeable, such as:
    • Variables in alphabetical order.
    • Constants at the end.
In our example, moving from \(3y + 4 - 4x\) to \(3y - 4x + 4\) arranges the variables and constant in a more common format.
This doesn鈥檛 change the meaning or value of the expression but makes calculations easier for comparison or matching against standard forms.

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Most popular questions from this chapter

Solve the equation. $$-5 x=-3 x+6$$

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Solve the equation. $$5(-4 x-8)=-9(-6 x+4)-4$$

The three sides of a triangle are consecutive odd integers. If the perimeter of the triangle is 165 inches, find the lengths of the sides of the triangle.

The width and length of a rectangle are consecutive odd integers. If the perimeter of the rectangle is 288 inches, find the width and length of the rectangle.
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