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Chapter 3: Problem 12

Solve each equation. $$3 x+1=21$$

Short Answer

Expert verified
\(x = \frac{20}{3}\)

Step by step solution

01

Isolate the term with the variable

Start by isolating the term with the variable on one side of the equation. Subtract 1 from both sides to remove the constant from the left side: \[3x + 1 - 1 = 21 - 1\] This simplifies to: \[3x = 20\]
02

Solve for the variable

Now, divide both sides of the equation by 3 to solve for \(x\):\[\frac{3x}{3} = \frac{20}{3}\] This results in: \[x = \frac{20}{3}\]

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Key Concepts

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

Solving Linear Equations
Linear equations are fundamental in algebra. They involve finding the value of a variable that makes the equation true. A linear equation is generally in the form \( ax + b = c \), where \( a \), \( b \), and \( c \) are constants, and \( x \) is the variable. The goal is to solve for the variable, meaning you want to find the number that \( x \) must be to make both sides of the equation equal. Here are some key points on why solving linear equations is important:
  • They represent real-world situations. For instance, calculating expenses or measuring distances.
  • Understanding basic algebra concepts increases problem-solving skills.
  • They form the foundation for more complex mathematical concepts, like quadratic equations or calculus.
Understanding how to manipulate these equations through operations such as addition, subtraction, multiplication, and division is crucial when tackling algebra problems.
Step-by-Step Solutions
Approaching algebraic problems becomes much easier with a methodical step-by-step approach. Breaking down the process into simple steps can help you methodically solve an equation and reduce the risk of mistakes. For example, consider the equation \( 3x + 1 = 21 \). By solving this step-by-step:
  • Start with simplifying the equation: Get rid of constants or extra terms by performing the opposite operation. Here, subtract \( 1 \) from both sides, resulting in \( 3x = 20 \).
  • Isolate the variable: Get \( x \) by itself by dividing both sides by \( 3 \), which gives \( x = \frac{20}{3} \).
This approach not only simplifies the solution process but also aids in better understanding and clarity of each stage in problem-solving.
Variable Isolation
Variable isolation is crucial for solving any algebraic equation. The core idea is to manipulate the equation so that the variable appears by itself on one side. This means you apply inverse operations, moving terms around until the variable is isolated. In the exercise \( 3x + 1 = 21 \), focusing on isolating \( x \) leads us to:
  • Subtract 1 from both sides: Remove the constant term by performing subtraction, yielding \( 3x = 20 \).
  • Divide by the coefficient: With \( 3x \) isolated, divide by \( 3 \) to isolate \( x \), resulting in \( x = \frac{20}{3} \).
Understanding the concept of variable isolation helps learners to plan and execute steps needed to simplify equations, an essential skill in algebra and beyond.

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Most popular questions from this chapter

Solve each equation. $$\frac{1}{2} x-\frac{4}{3}=-\frac{5}{6}$$

The sum of three-eighths of a number and five-sixths of the same number is 29 . Find the number.

Suppose that you are helping a friend with his homework, and he solves the equation \(19=14-x\) like this: $$ \begin{aligned} 19 &=14-x \\ 19+x &=14-x+x \end{aligned} $$ $$ \begin{aligned} 19+x &=14 \\ 19+x-19 &=14-19 \\ x &=-5 \end{aligned} $$ The solution set is \(\\{-5\\}\). Does he have the correct solution set? What would you say to him about his method of solving the equation?

Solve each equation. $$-2(x-6)=-(x-9)$$

Ike has some nickels and dimes amounting to \(\$ 2.90\). The number of dimes is one less than twice the number of nickels. How many coins of each kind does he have?
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