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Chapter 1: Problem 79

Identify each as an expression or an equation. \(x+y=9\)

Short Answer

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Equation

Step by step solution

01

Understand the Definition

An equation is a mathematical statement that asserts the equality of two expressions, typically including an equals sign (=).An expression is a combination of numbers, variables, and operators (such as +, -, *, /) that represents a specific value, and it does not include an equals sign.
02

Examine the Given Problem

Look at the given problem: \(x + y = 9\). Notice that it includes an equals sign (\(=\)).
03

Classify the Problem

Since the given problem includes an equals sign, it implies a relationship where two expressions are asserted to be equal. This fits the definition of an equation.

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Key Concepts

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

Algebraic Expression
An algebraic expression is a combination of numbers, variables, and mathematical operators like addition, subtraction, multiplication, and division. It represents a value but does not assert equality between two expressions. Consider the expression \(3x + 5\). Here, we have:
  • \(3x\): A term with a coefficient 3 and a variable \(x\)
  • \(5\): A constant term
  • \(+\): An operator indicating addition
Every algebraic expression aims to show a specific value depending on the value of the variable(s) it contains. For example, if \(x = 2\), then \(3x + 5\) equals \(3(2) + 5 = 6 + 5 = 11\). Unlike equations, expressions do not include an equals sign and therefore do not create a relationship of equality between two sides.
Mathematical Equation
A mathematical equation is a statement that asserts the equality of two expressions. Equations usually contain an equals sign (=) that separates the two expressions. For instance, consider the equation \(x + y = 9\):
  • \(x + y\) is an expression containing two variables: \(x\) and \(y\).
  • The equals sign (=) indicates that the two sides have the same value.
  • \(9\) is a constant that the left-hand expression equals.
When solving equations, the goal is often to find the values of the variables that make the equation true. For example, if \(x = 3\) and \(y = 6\), then \(x + y = 3 + 6 = 9\), satisfying the equation. Equations are fundamental in algebra because they allow us to model real-world situations and solve problems involving unknown values.
Variables and Constants
Variables and constants are fundamental components of expressions and equations.
  • Variables: Symbols that represent unknown or changeable values. Common variables include \(x\), \(y\), and \(z\). For example, in \(x + 2\), \(x\) is the variable.
  • Constants: Fixed values that do not change. In algebra, constants are usually numbers. For example, in the expression \(x + 2\), \(2\) is a constant.
Variables allow equations and expressions to generalize mathematical relationships, making them useful for a wide range of applications. Constants provide a specific value against which other values can be measured or added. Together, these elements form the backbone of algebraic problem-solving.

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Most popular questions from this chapter

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