Which Expression Is Equivalent To $1.2 Y + 4.5 - 3.4 Y - 6.3$?A. $5.5 Y - 10.2$ B. $ 4.6 Y − 1.8 4.6 Y - 1.8 4.6 Y − 1.8 [/tex] C. $-2.2 Y + 11.1$ D. $-2.2 Y - 1.8$
Introduction
Algebraic expressions are a fundamental concept in mathematics, and simplifying them is an essential skill for students and professionals alike. In this article, we will explore the process of simplifying algebraic expressions, with a focus on combining like terms and applying the order of operations.
What are Algebraic Expressions?
An algebraic expression is a mathematical statement that consists of variables, constants, and mathematical operations. It is a way of representing a value or a relationship between values using symbols and mathematical notation. Algebraic expressions can be simple or complex, and they can be used to solve equations, model real-world situations, and make predictions.
Simplifying Algebraic Expressions
Simplifying algebraic expressions involves combining like terms and applying the order of operations. Like terms are terms that have the same variable raised to the same power. For example, 2x and 4x are like terms because they both have the variable x raised to the power of 1.
Step 1: Combine Like Terms
To simplify an algebraic expression, we need to combine like terms. This involves adding or subtracting the coefficients of like terms. For example, if we have the expression 2x + 4x, we can combine the like terms by adding the coefficients: 2x + 4x = 6x.
Step 2: Apply the Order of Operations
The order of operations is a set of rules that tells us which operations to perform first when simplifying an algebraic expression. The order of operations is as follows:
- Parentheses: Evaluate expressions inside parentheses first.
- Exponents: Evaluate any exponential expressions next.
- Multiplication and Division: Evaluate any multiplication and division operations from left to right.
- Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.
Example: Simplifying an Algebraic Expression
Let's simplify the algebraic expression 1.2y + 4.5 - 3.4y - 6.3. To simplify this expression, we need to combine like terms and apply the order of operations.
Step 1: Combine Like Terms
The expression 1.2y + 4.5 - 3.4y - 6.3 has two like terms: 1.2y and -3.4y. We can combine these like terms by adding their coefficients: 1.2y - 3.4y = -2.2y.
Step 2: Apply the Order of Operations
Now that we have combined the like terms, we can apply the order of operations. We need to evaluate the constants first: 4.5 - 6.3 = -1.8.
Step 3: Simplify the Expression
Now that we have evaluated the constants, we can simplify the expression by combining the like terms and applying the order of operations: 1.2y + 4.5 - 3.4y - 6.3 = -2.2y - 1.8.
Conclusion
Simplifying algebraic expressions is an essential skill for students and professionals alike. By combining like terms and applying the order of operations, we can simplify complex expressions and make them easier to understand. In this article, we have explored the process of simplifying algebraic expressions, with a focus on combining like terms and applying the order of operations.
Which Expression is Equivalent to the Given Expression?
Now that we have simplified the algebraic expression 1.2y + 4.5 - 3.4y - 6.3, we can compare it to the given options to determine which expression is equivalent.
Option A: 5.5y - 10.2
This option is not equivalent to the given expression because it has a different coefficient for the variable y and a different constant.
Option B: 4.6y - 1.8
This option is not equivalent to the given expression because it has a different coefficient for the variable y and a different constant.
Option C: -2.2y + 11.1
This option is not equivalent to the given expression because it has a different constant.
Option D: -2.2y - 1.8
This option is equivalent to the given expression because it has the same coefficient for the variable y and the same constant.
Answer
The correct answer is Option D: -2.2y - 1.8. This option is equivalent to the given expression because it has the same coefficient for the variable y and the same constant.
Final Thoughts
Q: What is the order of operations?
A: The order of operations is a set of rules that tells us which operations to perform first when simplifying an algebraic expression. The order of operations is as follows:
- Parentheses: Evaluate expressions inside parentheses first.
- Exponents: Evaluate any exponential expressions next.
- Multiplication and Division: Evaluate any multiplication and division operations from left to right.
- Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.
Q: How do I combine like terms?
A: To combine like terms, you need to add or subtract the coefficients of like terms. For example, if you have the expression 2x + 4x, you can combine the like terms by adding the coefficients: 2x + 4x = 6x.
Q: What are like terms?
A: Like terms are terms that have the same variable raised to the same power. For example, 2x and 4x are like terms because they both have the variable x raised to the power of 1.
Q: How do I simplify an algebraic expression?
A: To simplify an algebraic expression, you need to combine like terms and apply the order of operations. Here's a step-by-step guide:
- Combine like terms.
- Apply the order of operations.
- Simplify the expression.
Q: What is the difference between a variable and a constant?
A: A variable is a symbol that represents a value that can change. For example, x is a variable. A constant is a value that does not change. For example, 5 is a constant.
Q: How do I evaluate an expression with parentheses?
A: To evaluate an expression with parentheses, you need to evaluate the expression inside the parentheses first. For example, if you have the expression 2(x + 3), you need to evaluate the expression inside the parentheses first: x + 3 = 4. Then, you can multiply 2 by 4: 2(4) = 8.
Q: What is the difference between an equation and an expression?
A: An equation is a statement that says two expressions are equal. For example, 2x + 3 = 5 is an equation. An expression is a mathematical statement that consists of variables, constants, and mathematical operations. For example, 2x + 3 is an expression.
Q: How do I simplify an expression with fractions?
A: To simplify an expression with fractions, you need to combine like terms and apply the order of operations. Here's a step-by-step guide:
- Combine like terms.
- Apply the order of operations.
- Simplify the expression.
Q: What is the difference between a linear expression and a quadratic expression?
A: A linear expression is an expression that has a variable raised to the power of 1. For example, 2x + 3 is a linear expression. A quadratic expression is an expression that has a variable raised to the power of 2. For example, x^2 + 2x + 3 is a quadratic expression.
Q: How do I simplify an expression with exponents?
A: To simplify an expression with exponents, you need to apply the order of operations. Here's a step-by-step guide:
- Evaluate any exponential expressions.
- Apply the order of operations.
- Simplify the expression.
Q: What is the difference between a positive exponent and a negative exponent?
A: A positive exponent is an exponent that is greater than 0. For example, x^2 is a positive exponent. A negative exponent is an exponent that is less than 0. For example, x^-2 is a negative exponent.
Q: How do I simplify an expression with absolute value?
A: To simplify an expression with absolute value, you need to apply the order of operations. Here's a step-by-step guide:
- Evaluate any absolute value expressions.
- Apply the order of operations.
- Simplify the expression.
Q: What is the difference between a rational expression and an irrational expression?
A: A rational expression is an expression that can be written as a fraction. For example, 2x + 3 is a rational expression. An irrational expression is an expression that cannot be written as a fraction. For example, x^2 + 2x + 3 is an irrational expression.
Q: How do I simplify an expression with radicals?
A: To simplify an expression with radicals, you need to apply the order of operations. Here's a step-by-step guide:
- Evaluate any radical expressions.
- Apply the order of operations.
- Simplify the expression.
Q: What is the difference between a real number and an imaginary number?
A: A real number is a number that can be written as a decimal or a fraction. For example, 2 is a real number. An imaginary number is a number that cannot be written as a decimal or a fraction. For example, i is an imaginary number.
Q: How do I simplify an expression with complex numbers?
A: To simplify an expression with complex numbers, you need to apply the order of operations. Here's a step-by-step guide:
- Evaluate any complex number expressions.
- Apply the order of operations.
- Simplify the expression.
Q: What is the difference between a polynomial expression and a non-polynomial expression?
A: A polynomial expression is an expression that has variables raised to non-negative integer powers. For example, 2x + 3 is a polynomial expression. A non-polynomial expression is an expression that has variables raised to non-integer powers. For example, x^2 + 2x + 3 is a non-polynomial expression.
Q: How do I simplify an expression with trigonometric functions?
A: To simplify an expression with trigonometric functions, you need to apply the order of operations. Here's a step-by-step guide:
- Evaluate any trigonometric function expressions.
- Apply the order of operations.
- Simplify the expression.
Q: What is the difference between a sine function and a cosine function?
A: A sine function is a trigonometric function that represents the ratio of the length of the side opposite a given angle to the length of the hypotenuse. For example, sin(x) is a sine function. A cosine function is a trigonometric function that represents the ratio of the length of the side adjacent to a given angle to the length of the hypotenuse. For example, cos(x) is a cosine function.
Q: How do I simplify an expression with logarithmic functions?
A: To simplify an expression with logarithmic functions, you need to apply the order of operations. Here's a step-by-step guide:
- Evaluate any logarithmic function expressions.
- Apply the order of operations.
- Simplify the expression.
Q: What is the difference between a natural logarithm and a common logarithm?
A: A natural logarithm is a logarithmic function that has a base of e. For example, ln(x) is a natural logarithm. A common logarithm is a logarithmic function that has a base of 10. For example, log(x) is a common logarithm.
Q: How do I simplify an expression with exponential functions?
A: To simplify an expression with exponential functions, you need to apply the order of operations. Here's a step-by-step guide:
- Evaluate any exponential function expressions.
- Apply the order of operations.
- Simplify the expression.
Q: What is the difference between a base-10 exponential function and a base-e exponential function?
A: A base-10 exponential function is an exponential function that has a base of 10. For example, 10^x is a base-10 exponential function. A base-e exponential function is an exponential function that has a base of e. For example, e^x is a base-e exponential function.
Q: How do I simplify an expression with absolute value functions?
A: To simplify an expression with absolute value functions, you need to apply the order of operations. Here's a step-by-step guide:
- Evaluate any absolute value function expressions.
- Apply the order of operations.
- Simplify the expression.
Q: What is the difference between a positive absolute value function and a negative absolute value function?
A: A positive absolute value function is an absolute value function that has a positive value. For example, |x| is a positive absolute value function. A negative absolute value function is an absolute value function that has a negative value. For example, -|x| is a negative absolute value function.
Q: How do I simplify an expression with rational functions?
A: To simplify an expression with rational functions, you need to apply the order of operations. Here's a step-by-step guide:
- Evaluate any rational function expressions.
- Apply the order of operations.
- Simplify the expression.
Q: What is the difference between a rational function and an irrational function?
A: A rational function is a function that can be written as a fraction. For example, 2x + 3 is a rational function. An irrational function is a function that cannot be written as a fraction