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Chapter 2: Problem 16

Simplify each expression as completely as possible. $$5(r+4 s)-3(3 s-2 r)$$

Short Answer

Expert verified
11r + 11s

Step by step solution

01

- Distribute the Constants

Distribute the constants outside the parentheses to each term inside the parentheses. This means multiplying each term inside the parentheses by the constant outside.The expression is: \[ 5(r+4s) - 3(3s - 2r) \]This gives:\[ 5r + 20s - 3 \times 3s - 3 \times (-2r) \]
02

- Simplify Each Distributed Term

Simplify the distributed terms by performing the multiplication.\[ 5r + 20s - 9s + 6r \]
03

- Combine Like Terms

Combine the like terms to simplify the expression. Like terms are those terms that have the same variable raised to the same power.Combine the terms with 'r': \[ 5r + 6r = 11r \]Combine the terms with 's':\[ 20s - 9s = 11s \]Thus, the simplified expression is:\[ 11r + 11s \]

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Key Concepts

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

distributive property
When simplifying algebraic expressions, the distributive property is a crucial concept. This property involves multiplying a single term by each term inside a set of parentheses. So, if you have an expression like \(a (b + c)\), you'll distribute 'a' to both 'b' and 'c'.
In our specific exercise, we started with \[ 5(r + 4s) - 3(3s - 2r) \].
Using the distributive property, we distribute '5' to both 'r' and '4s', and '-3' to both '3s' and '-2r':
\[ 5r + 20s - 3 \times 3s - 3 \times (-2r) \]
This step lays the foundation for simplifying complex expressions by breaking them down into simpler parts.
combining like terms
After distributing the constants, the next step is to simplify by combining like terms. 'Like terms' are terms that have the same variable raised to the same power. For example, '5r' and '6r' are like terms, as are '20s' and '-9s'.
Looking at our exercise again, we had the intermediate result:
\[ 5r + 20s - 9s + 6r \]
First, we combined the 'r' terms:
\[ 5r + 6r = 11r \]
Then, we combined the 's' terms:
\[ 20s - 9s = 11s \]
Combining like terms simplifies the expression and brings us closer to the final, simplified form:
\[ 11r + 11s \]
basic algebra operations
Basic algebra operations such as addition, subtraction, multiplication, and division are the foundation of solving algebra problems. First, ensure all distributions and operations follow the order of operations (PEMDAS).
In our example:
\[ 5(r + 4s) - 3(3s - 2r) \]
We distributed the constants:\br>\[ 5r + 20s - 9s + 6r \]
Then, using addition and subtraction, we simplified the expression by combining the like terms:
\[ 11r + 11s \]
Practicing these basic operations makes it easier to handle more complex algebraic expressions in the future.

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Most popular questions from this chapter

Evaluate the given expression for \(x=2, y=-3,\) and \(z=-4.\) $$x^{2}+y^{2}+z^{2}$$

Evaluate the given expression for \(x=2, y=-3,\) and \(z=-4.\) $$\frac{x y z}{z-x}$$

Make up an example of an expression that consists of three terms, one of which has one factor, one of which has two factors, and one of which has three factors.

Explain what is wrong with each of the following: (a) \(3 \cdot 2^{2} \stackrel{?}{=} 6^{2} \stackrel{?}{=} 36\) (b) \(3 x^{4} \stackrel{?}{=} 3 x \cdot 3 x \cdot 3 x \cdot 3 x\) (c) \(5^{4} \cdot 2 \stackrel{?}{=} 20 \cdot 2 \stackrel{?}{=}40\) (d) \(\left(3 x^{3}\right)^{2} \stackrel{?}{=} 6 x^{6}\) (e) \(-3^{4} \stackrel{?}{=}(-3) \cdot(-3) \cdot(-3)(-3) \stackrel{?}{=} 81\) (f) \(x^{4}+x^{5} \stackrel{?}{=} x^{9}\) (g) \(x^{2} \cdot x^{7} \stackrel{?}{=} x^{14}\) (h) \(\left(x^{2}\right)^{4} \stackrel{?}{=} x^{6}\)

Evaluate the given expression for \(x=2, y=-3,\) and \(z=-4.\) $$\frac{2}{x}-\frac{y}{3}+\frac{z}{2}$$
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