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Chapter 3: Problem 63

Simplify each expression. $$ -(4 x-5)+5 $$

Short Answer

Expert verified
-4x + 10

Step by step solution

01

Distribute the Negative Sign

Distribute the negative sign through the parentheses in the expression, transforming each term inside: \( -(4x - 5) \) becomes \(-4x + 5\). This step results in the expression \(-4x + 5 + 5\).
02

Combine Like Terms

Add the numbers together. You have two constant terms in the expression: \(5\) and \(5\). Combine these to simplify the expression further: \(-4x + 5 + 5 = -4x + 10\).

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Key Concepts

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

Distributing Negative Signs
When simplifying algebraic expressions, one crucial step is distributing negative signs. A negative sign outside parentheses affects each term inside.
This process involves multiplying each term within the parentheses by -1. Consider the expression \(-(4x - 5)\). The negative sign will alter the signs of all terms:
  • -4x becomes -1 times 4x, resulting in \(-4x\).
  • 5 becomes -1 times -5, resulting in \(+5\).
The alteration gives us a transformed expression: \(-4x + 5\). Remember, changing the sign is like flipping the sign of each term.
It's essential to pay attention to these changes as they can significantly affect the outcome of your simplification!
Combining Like Terms
Combining like terms is another fundamental step in simplifying expressions. Like terms are terms in an expression that contain the same variable raised to the same power.
For example, in the expression \(-4x + 5 + 5\), the like terms happen to be the constant numbers, namely the 5s. Combining them is simply adding them together.
  • The expression \(5 + 5\) equals \(10\).
The variables themselves, like \(-4x\), have no like terms here, so they remain unchanged.
By adding like constants together, the expression simplifies to \(-4x + 10\). Combining like terms helps you clean up the expression and make it more straightforward.
Algebraic Expressions
In math, an algebraic expression is a combination of numbers, variables, and arithmetic operators. Simplifying these expressions involves applying mathematical rules to achieve a more compact form.
The process generally includes steps like distributing signs, combining like terms, and occasionally factoring.
  • An "expression" might include elements like \(4x\), \(-5\), or constants like \(5\).
  • Operators such as addition and subtraction connect these elements.
The example \(-4x + 10\) started as a longer expression, \(-(4x - 5) + 5\), showcasing how algebraic expressions can be simplified into fewer terms.
Through practice, these steps become intuitive, enabling smoother transitions from complex forms to simplified ones.

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