Which Equation Is Equivalent To $4.5 = 2a$?A. $a = 2.5$ B. \$a = 2.25$[/tex\] C. $4.5a = 2$ D. $a + 2 = 4.5$
Introduction to Algebraic Equations
In algebra, equations are used to represent relationships between variables. These relationships can be expressed in various forms, and understanding the different types of equations is crucial for solving mathematical problems. In this article, we will explore which equation is equivalent to the given equation $4.5 = 2a$.
Understanding the Given Equation
The given equation is $4.5 = 2a$. This equation states that the value of $4.5$ is equal to twice the value of $a$. To find the value of $a$, we need to isolate $a$ on one side of the equation.
Isolating the Variable
To isolate $a$, we can divide both sides of the equation by $2$. This will cancel out the $2$ on the right-hand side, leaving us with the value of $a$.
Solving the Equation
Let's solve the equation step by step:
Divide both sides by $2$:
Simplify the equation:
Analyzing the Options
Now that we have solved the equation, let's analyze the options:
A. $a = 2.5$
This option is incorrect because the value of $a$ is $2.25$, not $2.5$.
B. $a = 2.25$
This option is correct because the value of $a$ is indeed $2.25$.
C. $4.5a = 2$
This option is incorrect because the equation $4.5a = 2$ is not equivalent to the given equation $4.5 = 2a$.
D. $a + 2 = 4.5$
This option is incorrect because the equation $a + 2 = 4.5$ is not equivalent to the given equation $4.5 = 2a$.
Conclusion
In conclusion, the correct answer is option B: $a = 2.25$. This option is equivalent to the given equation $4.5 = 2a$.
Frequently Asked Questions
- What is the value of $a$ in the equation $4.5 = 2a$?
- How do I isolate the variable $a$ in the equation $4.5 = 2a$?
- What is the equivalent equation to $4.5 = 2a$?
Step-by-Step Solution
- Start with the given equation: $4.5 = 2a$
- Divide both sides of the equation by $2$: $\frac{4.5}{2} = \frac{2a}{2}$
- Simplify the equation: $2.25 = a$
- The value of $a$ is $2.25$.
Final Answer
The final answer is option B: $a = 2.25$.
Introduction
Algebraic equations are a fundamental concept in mathematics, and understanding them is crucial for solving mathematical problems. In this article, we will address some frequently asked questions about algebraic equations, including how to isolate variables, solve equations, and determine equivalent equations.
Q&A
Q: What is an algebraic equation?
A: An algebraic equation is a mathematical statement that expresses the relationship between variables and constants. It consists of an equal sign (=) that separates two expressions, with the variable(s) on one side and the constant(s) on the other.
Q: How do I isolate a variable in an algebraic equation?
A: To isolate a variable, you need to get it alone on one side of the equation. This can be done by performing operations such as addition, subtraction, multiplication, or division to both sides of the equation. For example, if you have the equation $x + 3 = 7$, you can isolate $x$ by subtracting $3$ from both sides: $x = 7 - 3$.
Q: What is the difference between a linear equation and a quadratic equation?
A: A linear equation is an equation in which the highest power of the variable is $1$. For example, $2x + 3 = 5$ is a linear equation. A quadratic equation, on the other hand, is an equation in which the highest power of the variable is $2$. For example, $x^2 + 4x + 4 = 0$ is a quadratic equation.
Q: How do I solve a linear equation?
A: To solve a linear equation, you need to isolate the variable. This can be done by performing operations such as addition, subtraction, multiplication, or division to both sides of the equation. For example, if you have the equation $2x + 3 = 5$, you can solve it by subtracting $3$ from both sides: $2x = 5 - 3$, and then dividing both sides by $2$: $x = \frac{5 - 3}{2}$.
Q: What is the order of operations in algebra?
A: The order of operations in algebra is a set of rules that tells you which operations to perform first when solving an equation. The order of operations is:
- 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 determine if two equations are equivalent?
A: Two equations are equivalent if they have the same solution. To determine if two equations are equivalent, you can try to solve one of the equations and see if the solution satisfies the other equation. Alternatively, you can try to manipulate one of the equations to make it look like the other equation.
Q: What is the difference between a dependent and an independent variable?
A: In an algebraic equation, the dependent variable is the variable that is being solved for, while the independent variable is the variable that is being manipulated to solve for the dependent variable. For example, in the equation $y = 2x + 3$, $y$ is the dependent variable and $x$ is the independent variable.
Conclusion
In conclusion, algebraic equations are a fundamental concept in mathematics, and understanding them is crucial for solving mathematical problems. By following the steps outlined in this article, you can isolate variables, solve equations, and determine equivalent equations.
Frequently Asked Questions (FAQs) About Algebraic Equations
- What is an algebraic equation?
- How do I isolate a variable in an algebraic equation?
- What is the difference between a linear equation and a quadratic equation?
- How do I solve a linear equation?
- What is the order of operations in algebra?
- How do I determine if two equations are equivalent?
- What is the difference between a dependent and an independent variable?
Step-by-Step Solution
- Start with the given equation.
- Identify the variable and the constant.
- Perform operations to isolate the variable.
- Check if the equation is linear or quadratic.
- Solve the equation using the appropriate method.
- Check if the solution satisfies the original equation.
Final Answer
The final answer is that algebraic equations are a fundamental concept in mathematics, and understanding them is crucial for solving mathematical problems. By following the steps outlined in this article, you can isolate variables, solve equations, and determine equivalent equations.