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Chapter 11: Problem 51

Solve. $$ 5 r-(2 r+8)=16 $$

Short Answer

Expert verified
The solution is \(r = 8\).

Step by step solution

01

Distribute and Simplify

First, distribute any factors and simplify inside the parentheses. Start with: \(5r - (2r + 8) = 16\) This becomes: \(5r - 2r - 8 = 16\)
02

Combine Like Terms

Combine the like terms on the left side. \(5r - 2r\) combines to \(3r\), resulting in: \(3r - 8 = 16\)
03

Isolate the Variable Term

To get the term with the variable by itself, add 8 to both sides of the equation: \(3r - 8 + 8 = 16 + 8\), which simplifies to: \(3r = 24\)
04

Solve for the Variable

Finally, divide both sides by 3 to solve for the variable \(r\): \(\frac{3r}{3} = \frac{24}{3}\), which gives: \(r = 8\)

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Key Concepts

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

Algebraic Expressions
An algebraic expression is a combination of numbers, variables, and operations, such as addition and multiplication. In our example, the expression is 5r - (2r + 8). Here, 'r' is the variable, and the numbers 5, 2, and 8 are constants. The operations involved are multiplication (5r and 2r) and subtraction. Understanding how to manipulate and simplify such expressions is key to solving equations.
Combining Like Terms
Like terms are terms that have the same variables raised to the same power. In the expression 5r - 2r - 8, 5r and 2r are like terms because they both contain the variable 'r'. When we combine them: 5r - 2r, we get 3r. This step simplifies the equation to 3r - 8 = 16. Combining like terms helps in reducing the equation, making it simpler to solve. Always look out for like terms to combine, as it brings clarity to the equation.
Isolating Variables
Isolating the variable means getting the variable 'r' by itself on one side of the equation. In the simplified equation 3r - 8 = 16, we need to get rid of the constant term (-8). We do this by adding 8 to both sides: 3r - 8 + 8 = 16 + 8, which simplifies to 3r = 24. Next, we divide both sides by 3: \[ \frac{3r}{3} = \frac{24}{3} \], to get r = 8. This process of isolation is crucial, as it transforms the equation into a form where the value of the variable can be easily determined.

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