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Chapter 2: Problem 1

Fill in the blanks. \(3 x+8=10\) is an example of a linear _____ in one variable.

Short Answer

Expert verified
The answer is 'equation'.

Step by step solution

01

Identify the form of the equation

The given equation is in the form of \( ax + b = c \), where \( a = 3 \), \( b = 8 \), and \( c = 10 \). This is a standard form for a linear equation in one variable.
02

Understand the characteristics of linear equations

A linear equation in one variable will always produce a straight line when graphed on a coordinate plane. In this case, the equation involves only one variable, \( x \). It is linear because it does not include exponents greater than one, and it involves operations that can be represented as a straight line.
03

Fill in the blank using the characteristics

Since the equation involves one variable (\( x \)) and follows the structure of a straight-line equation, the blank should be filled with the word "equation". So, it reads "\(3 x+8=10\) is an example of a linear equation in one variable."

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Key Concepts

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

Variables in Algebra
Variables are the backbone of algebraic expressions and equations. They are symbols, often letters, used to represent numbers in equations or expressions. This use of letters allows for generalization and manipulation of numbers in a variety of mathematical situations.
For example, in the equation \(3x + 8 = 10\), "\(x\)" is the variable. It stands in place of a number that makes the equation true. By solving the equation, we discover the value of the variable that balances both sides of the equation.
  • Variables can represent unknown quantities or a set of numbers.
  • Common letters used for variables include \(x\), \(y\), and \(z\).
  • They allow for the formation of general rules in formulas used across various problems.
Understanding the role of a variable is crucial in unraveling and solving equations effectively.
Solving Equations
Solving equations involves finding the value of the variable that makes the equation true. This means manipulating the algebraic expression step by step until you isolate the variable. For the linear equation \(3x + 8 = 10\), solving the equation involves a clear sequence of steps:
  • Step 1: Subtract 8 from both sides to remove the constant term: \(3x + 8 - 8 = 10 - 8\). This simplifies to \(3x = 2\).
  • Step 2: Divide both sides by 3 to solve for \(x\): \(x = \frac{2}{3}\).
By following these steps, you isolate \(x\) on one side. This process is a fundamental skill in algebra that can be applied to more complex equations as well.
Algebraic Expressions
Algebraic expressions are combinations of numbers, variables, and operators (such as addition or multiplication). They do not include an equality sign, unlike equations. In our example, the algebraic expression part before it becomes an equation is \(3x + 8\).
An algebraic expression can be as simple as \(x + 5\) or as complex as \(2x^2 + 3x + 7\).
  • Expressions can consist of just one term (a monomial) or multiple terms (polynomial).
  • They are fundamental in creating equations by equating them to a value or another expression.
  • Simplifying algebraic expressions involves consolidating like terms and applying distributive laws.
Grasping the concept of algebraic expressions is important because it forms the basis upon which equations are built and solved.

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