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Chapter 1: Problem 17

Simplify the given algebraic expressions. $$2-3-(4-5 a)$$

Short Answer

Expert verified
The simplified expression is \(5a - 5\).

Step by step solution

01

Remove Parentheses

The first task is to simplify the expression by removing the parentheses. The outer expression is \(2 - 3 - (4 - 5a)\). We start by distributing the negative sign to the terms inside the parenthesis, changing \(4 - 5a\) to \(-4 + 5a\). This leads to the expression: \(2 - 3 - 4 + 5a\).
02

Simplify Constants

After removing parentheses, simplify the constants. Combine \(2, -3, \) and \(-4\). First, compute \(2 - 3 = -1\), then \(-1 - 4 = -5\). This leaves us with \(-5 + 5a\).
03

Final Simplified Expression

Now, arrange the terms to present the fully simplified expression. The constants and variable terms are separated into \(-5\) and \(5a\). The simplified form of the original expression is \(5a - 5\).

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Key Concepts

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

Simplifying Expressions
When we talk about simplifying expressions in algebra, we're essentially streamlining a mathematical problem to make it easier to handle. Imagine you're tidying up a messy room; similarly, simplifying involves getting rid of any unnecessary parts and organizing the rest. This process requires a few strategic maneuvers:
  • Removing any redundant parentheses
  • Utilizing properties like the distributive property
  • Combining like terms to merge similar entities together
Each step helps clear up the expression, making the math easier to digest and solve. The goal is to end with a clean and neat algebraic expression.
Removing Parentheses
Parentheses in math serve a lot like packaging鈥攖hey group certain parts of an equation together. But sometimes, to simplify an expression, these groups need to be undone. To remove parentheses, follow these steps:
  • Identify the parentheses in the expression.
  • Look out for any operators, like a minus sign, directly before them.
  • Apply any necessary changes. For example, if a negative sign precedes the parenthesis, change the signs of all terms inside when you "unpack" it.
For example, in the expression \(2 - 3 - (4 - 5a)\), when removing the parentheses, consider the negative sign in front. This changes the signs of the terms inside, transforming \(4 - 5a\) into \(-4 + 5a\). So, this step helps us reach the simpler expression \(2 - 3 - 4 + 5a\).
Distributive Property
The distributive property is a handy tool in algebra that lets you multiply a number across the terms inside a parenthesis. Generally, this looks like \(a(b + c) = ab + ac\). It allows us to "distribute" a factor across terms inside parentheses:
  • Multiply the outer term by each term inside
  • After distribution, remove the parentheses
In our original expression, while the direct application of distribution wasn't needed for all terms, the negative sign in \(-(4 - 5a)\) acts similarly. This minus sign multiplies both \(4\) and \(-5a\) by \(-1\), resulting in \(-4 + 5a\). This step refines the expression so we can more easily combine and simplify terms.
Combining Like Terms
Combining like terms is the algebraic version of merging similar items. If you have a stack of like items, you group them together to simplify inventory.
In algebra, terms with the same variable part or constants are combined:
  • Identify the terms that are "like"鈥攎eaning they have the same variables and powers
  • Add or subtract these like terms as applicable
In the expression we derived \(2 - 3 - 4 + 5a\), the numbers \(2\), \(-3\), and \(-4\) are constants, which we can combine:
- First, \(2 - 3 = -1\)
- Then, \(-1 - 4 = -5\)
This leaves us with \(-5\), which cannot merge with the \(5a\), because they're not alike. So our final result is the clean and simplified \(5a - 5\).

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Most popular questions from this chapter

Simplify the given algebraic expressions. For the following expressions, subtract the third from the sum of the first two: \(3 a^{2}+b-c^{3}, 2 c^{3}-2 b-a^{2}, 4 c^{3}-4 b+3\).

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Set up an appropriate equation and solve. Data are accurate to two sig. digits unless greater accuracy is given. A ski lift takes a skier up a slope at \(50 \mathrm{m} / \mathrm{min}\). The skier then skis down the slope at \(150 \mathrm{m} / \mathrm{min} .\) If a round trip takes \(24 \mathrm{min}\), how long is the slope?

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Solve the given equations. All numbers are approximate. $$\frac{x}{2.0}=\frac{17}{6.0}$$
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