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Chapter 3: Problem 1

Identify each as an expression or an equation. $$7 t-2=11$$

Short Answer

Expert verified
The given expression is \(7t - 2 = 11\), which contains an equal sign, showing the relationship between two expressions. Therefore, it is an equation.

Step by step solution

01

Definition of Expression and Equation

An expression is a mathematical phrase composed of numbers, variables, and mathematical operations, but without an equal sign. In contrast, an equation is a mathematical statement that shows the relationship between two expressions using an equal sign.
02

Examine the given expression

Now, let's look at the given expression: \(7t - 2 = 11\)
03

Determine whether it is an expression or an equation

Since the given expression contains an equal sign, which shows the relationship between two expressions, it is an equation.

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Key Concepts

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

mathematical expressions
When dealing with mathematical expressions, it's essential to understand that these are the building blocks of algebra. A mathematical expression is an arrangement involving numbers, variables, and operations like addition, subtraction, multiplication, and division. Here are some key points:
  • No equal sign: Unlike equations, expressions do not have an equal sign. This means they don't equate to anything, they are just phrases.
  • They can be simple or complex: An expression can be as simple as a single number or variable, like 5 or \(x\), or more complex like \(3x + 7\).
  • Used for evaluation: Expressions can be evaluated if the values of the variables are known. For example, if \(x = 2\), then the expression \(3x + 7\) would evaluate to 13.
Expressions are versatile and can be transformed through algebraic operations, but they inherently represent a single value or quantity.
equation vs expression
Often, students find themselves confused between expressions and equations. Understanding the difference is crucial for problem-solving in algebra. Let's explore:

  • Equations: These are statements showing that two expressions are equal. They always include an equal sign, for instance, \(7t - 2 = 11\). This means that the expression on the left is equivalent to the expression on the right.
  • Expressions: As mentioned earlier, they are like phrases that do not claim equality with something else. They could look like \(3x + 2\) or \(a^2 - b^2\), standing on their own.
Why does this matter? Knowing whether you're working with an equation or expression will determine the kind of operations you can perform. With equations, you're often tasked with finding the value of the variable that makes the statement true. With expressions, you're usually simplifying or evaluating.
algebraic operations
Algebraic operations are the fundamental processes used to manipulate expressions and solve equations. They are the basic tools of algebra and include addition, subtraction, multiplication, and division. Here's how each is typically used:
  • Addition and subtraction: These operations are used to combine or separate quantities. For instance, \(x + 3 - 2\) is simplified by combining the numbers to \(x + 1\).
  • Multiplication and division: These operations involve scaling expressions. For instance, multiplying \(x\) by 2 gives \(2x\), while dividing \(6x\) by 3 results in \(2x\).
  • Simplifying expressions: Algebraic operations help in rearranging the expression into a simpler form. This involves combining like terms and factoring.
  • Solving equations: These operations aid in isolating the variable to find solutions. For instance, in the equation \(7t - 2 = 11\), adding 2 to both sides and then dividing by 7 will solve for \(t\).
Mastering these operations is crucial for anyone who wants to excel in algebra, as they are repeated processes applied to a wide range of problems.

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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." Tamara invests some money in three different accounts. She puts some of it in a CD earning \(3 \%\) simple interest and twice as much in an IRA paying \(4 \%\) simple interest. She also decides to invest \(\$ 1000\) more than what she's invested in the CD into a mutual fund earning \(5 \%\) simple interest. Determine how much money Tamara invested in each account if she earned \(\$ 290\) in interest after 1 year.

Solve for \(d\). a) \(8 d-7=17\) b) \(k d-a=z\)

Solve using the five "Steps for Solving Applied Problems." See Example 9. At noon, a truck and a car leave the same intersection traveling in the same direction. The truck is traveling at \(30 \mathrm{mph},\) and the car is traveling at \(42 \mathrm{mph} .\) At what time will they be 9 mi apart?

Solve using the five "Steps for Solving Applied Problems." See Examples \(7-9\) Nine less than a number is eleven. Find the number.

Objectives 2 and 3 Mixed Exercises Solve using the five "Steps for Solving Applied Problems." On his 21 st birthday, Jerry received \(\$ 20,000\) from a trust fund. He invested some of the money in an account earning \(5 \%\) simple interest and put the rest in a high-risk investment paying \(9 \%\) simple interest. After 1 year, he earned a total of \(\$ 1560\) in interest. How much did Jerry invest in each account?
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