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Chapter 3: Problem 50

Solve and check each equation. $$-3 r+8=8$$

Short Answer

Expert verified
The solution to the equation \(-3r + 8 = 8\) is \(r = 0\). This value of \(r\) satisfies the equation, as \(-3(0) + 8 = 8\).

Step by step solution

01

Rewrite the equation

First, let's rewrite the given equation: \[-3r + 8 = 8\]
02

Isolate the term with the variable

Now, we'll isolate the term with the variable, which is \(-3r\). To do that, we need to subtract 8 from both sides of the equation. \[-3r + 8 - 8 = 8 - 8\]
03

Simplify the equation

Now, let's simplify the equation: \[-3r = 0\]
04

Solve for the variable

To solve for the variable \(r\), we need to divide both sides of the equation by \(-3\). \[\frac{-3r}{-3} = \frac{0}{-3}\]
05

Find the value of the variable

Now, we'll find the value of the variable \(r\): \[r = 0\]
06

Check the solution

Finally, let's check if the obtained value of \(r\) satisfies the given equation. To do that, we'll substitute the value of \(r\) back into the equation: \[-3(0) + 8 = 8\] \[0 + 8 = 8\] \[8 = 8\] Since the equation is true for \(r = 0\), we can conclude that our solution is correct.

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Key Concepts

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

Understanding Algebra Concepts
Algebra is like a mathematical language, using symbols and numbers to express relationships. It helps us solve problems by figuring out unknown values. In our given equation, \(-3r + 8 = 8\), we are dealing with a **linear equation**. This means that the equation involves only a constant term and a single variable raised to the first power. Linear equations are one of the simplest forms and a crucial part of algebra, serving as a foundation for more complicated expressions.

Solving linear equations involves a few basic steps:
  • Recognizing the equation type.
  • Isolating the variable term.
  • Simplifying the equation.
  • Finding the value that makes the equation true.
Each step requires a bit of practice and understanding of arithmetic operations to manipulate the equation correctly.
Working with Variables in Equations
In algebra, **variables** are symbols used to represent unknown numbers. They are very versatile and can represent almost any value. Think of them as blanks in a sentence that need to be filled to make things meaningful. In our equation \(-3r + 8 = 8\), **r** is the variable. Our goal is to find the value of **r** that makes this equation true.

When working with variables:
  • Understand that they are placeholders for unknown values.
  • Always aim to isolate the variable on one side of the equation to solve for it.
  • Perform the same operation on both sides to maintain balance in the equation.

Here, we start by isolating the variable, **r**, by performing operations such as addition, subtraction, multiplication, or division until we find the desired value.
Checking Solutions to Equations
After finding the value of a variable, it is important to verify the solution by substituting it back into the original equation. This step confirms that the value indeed satisfies the equation. In our example, once we found \(r = 0\), the checking process is straightforward:

  • Substitute the value of **r** back into the equation: \(-3(0) + 8 = 8\).
  • Simplify both sides to ensure they are equal: \(0 + 8 = 8\).
  • Conclude that since both sides equal, the solution is correct.

Checking solutions not only confirms correctness but also helps solidify understanding of the variables and operations involved. It's a crucial step, especially when dealing with more complex equations in algebra.

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Most popular questions from this chapter

Objectives 2 and 3 Mixed Exercises Solve using the five "Steps for Solving Applied Problems." In August 2007 , there were 2600 entering freshmen at a state university. This is 4\% higher than the number of freshmen entering the school in August of 2006 How many freshmen were enrolled in August \(2006 ?\)

Solve each equation by first clearing fractions or decimals. $$\frac{1}{6} x+\frac{5}{4}=\frac{1}{2} x-\frac{5}{12}$$

Objectives 2 and 3 Mixed Exercises Solve using the five "Steps for Solving Applied Problems." Zoe's current salary is \(\$ 40,144\). This is \(4 \%\) higher than last year's salary. What was Zoe's salary last year?

Solve each equation by first clearing fractions or decimals. $$\frac{3}{4} n+\frac{1}{2}=\frac{1}{2} n+\frac{1}{4}$$

Solve using the five "Steps for Solving Applied Problems." See Examples \(7-9\) One-third of a number increased by four is one. Find the number.
See all solutions

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