Which Equation Can Be Used To Represent three Minus The Difference Of A Number And One Equals One-half Of The Difference Of Three Times The Same Number And Four?A. $3 - (1 - N) = \frac{1}{2}(4 - 3n$\]B. $3 - (n - 1) = \frac{1}{2}(3n -
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Understanding the Problem
In this article, we will delve into the world of algebraic equations and explore how to represent a given statement using a mathematical equation. The statement in question is: "three minus the difference of a number and one equals one-half of the difference of three times the same number and four." This statement can be broken down into a mathematical equation, which we will attempt to solve.
Breaking Down the Statement
To begin, let's break down the statement into its individual components. We have:
- Three minus the difference of a number and one
- Equals one-half of the difference of three times the same number and four
Representing the Statement as an Equation
Let's represent the statement as an equation. We can start by assigning a variable to the unknown number. Let's call this variable "n".
The statement can be rewritten as:
3 - (1 - n) = 1/2(3n - 4)
Simplifying the Equation
To simplify the equation, we can start by evaluating the expressions inside the parentheses.
3 - (1 - n) = 3 - 1 + n = 2 + n
Now, let's simplify the right-hand side of the equation:
1/2(3n - 4) = 1.5n - 2
Equating the Two Expressions
Now that we have simplified both sides of the equation, we can equate the two expressions:
2 + n = 1.5n - 2
Solving for n
To solve for n, we can start by adding 2 to both sides of the equation:
2 + n + 2 = 1.5n - 2 + 2 = 1.5n
Next, we can subtract 1.5n from both sides of the equation:
2 + n - 1.5n = 1.5n - 1.5n = 0.5n + 2
Now, we can subtract 2 from both sides of the equation:
0.5n + 2 - 2 = 0.5n = 0.5n
Finally, we can divide both sides of the equation by 0.5:
(0.5n) / 0.5 = 0.5n / 0.5 = n
Conclusion
In conclusion, the equation that represents the statement "three minus the difference of a number and one equals one-half of the difference of three times the same number and four" is:
3 - (1 - n) = 1/2(3n - 4)
This equation can be simplified to:
2 + n = 1.5n - 2
Solving for n, we get:
n = 4
Therefore, the correct answer is:
A.
Discussion
This problem requires a strong understanding of algebraic equations and the ability to simplify complex expressions. The key to solving this problem is to break down the statement into its individual components and represent it as an equation. From there, we can simplify the equation and solve for the unknown variable.
Real-World Applications
This type of problem has real-world applications in fields such as physics, engineering, and economics. For example, in physics, we may need to solve equations to describe the motion of objects or the behavior of electrical circuits. In engineering, we may need to solve equations to design and optimize systems. In economics, we may need to solve equations to model the behavior of markets or the impact of policy changes.
Tips and Tricks
When solving algebraic equations, it's essential to follow the order of operations (PEMDAS):
- 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.
By following this order of operations, we can ensure that we simplify the equation correctly and avoid any errors.
Common Mistakes
When solving algebraic equations, it's easy to make mistakes. Some common mistakes include:
- Not following the order of operations (PEMDAS)
- Not simplifying the equation correctly
- Not solving for the unknown variable
- Not checking the solution for validity
To avoid these mistakes, it's essential to take your time and carefully evaluate each step of the solution.
Conclusion
In conclusion, solving algebraic equations requires a strong understanding of algebraic concepts and the ability to simplify complex expressions. By following the order of operations (PEMDAS) and carefully evaluating each step of the solution, we can ensure that we solve the equation correctly and avoid any errors. This type of problem has real-world applications in fields such as physics, engineering, and economics, and is an essential skill for anyone working in these fields.
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Q: What is an algebraic equation?
A: An algebraic equation is a mathematical statement that contains one or more variables, which are symbols that represent unknown values. The equation is typically written in the form of an equation, where the variable(s) are set equal to a value or expression.
Q: How do I simplify an algebraic equation?
A: To simplify an algebraic equation, you can follow the order of operations (PEMDAS):
- 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: 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(s) 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(s) 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 can follow these steps:
- Isolate the variable(s) on one side of the equation.
- Use inverse operations to eliminate any coefficients or constants.
- Simplify the equation to find the value of the variable(s).
Q: How do I solve a quadratic equation?
A: To solve a quadratic equation, you can follow these steps:
- Factor the equation, if possible.
- Use the quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a
- Simplify the equation to find the value of the variable(s).
Q: What is the quadratic formula?
A: The quadratic formula is a mathematical formula that is used to solve quadratic equations. The formula is:
x = (-b ± √(b^2 - 4ac)) / 2a
where a, b, and c are the coefficients of the quadratic equation.
Q: How do I graph a linear equation?
A: To graph a linear equation, you can follow these steps:
- Find the x-intercept and y-intercept of the equation.
- Plot the x-intercept and y-intercept on a coordinate plane.
- Draw a line through the two points to graph the equation.
Q: How do I graph a quadratic equation?
A: To graph a quadratic equation, you can follow these steps:
- Find the x-intercepts and y-intercepts of the equation.
- Plot the x-intercepts and y-intercepts on a coordinate plane.
- Draw a parabola through the two points to graph the equation.
Q: What is the difference between a function and a relation?
A: A function is a relation in which each input value corresponds to exactly one output value. A relation, on the other hand, is a set of ordered pairs that may or may not have a one-to-one correspondence between the input and output values.
Q: How do I determine if a relation is a function?
A: To determine if a relation is a function, you can follow these steps:
- Check if each input value corresponds to exactly one output value.
- If each input value corresponds to exactly one output value, then the relation is a function.
Q: What is the domain of a function?
A: The domain of a function is the set of all input values for which the function is defined.
Q: What is the range of a function?
A: The range of a function is the set of all output values for which the function is defined.
Q: How do I find the domain and range of a function?
A: To find the domain and range of a function, you can follow these steps:
- Check the function for any restrictions on the input values.
- Determine the set of all input values for which the function is defined.
- Determine the set of all output values for which the function is defined.
Q: What is the difference between a linear function and a quadratic function?
A: A linear function is a function in which the highest power of the variable(s) is 1. A quadratic function, on the other hand, is a function in which the highest power of the variable(s) is 2.
Q: How do I graph a linear function?
A: To graph a linear function, you can follow these steps:
- Find the x-intercept and y-intercept of the function.
- Plot the x-intercept and y-intercept on a coordinate plane.
- Draw a line through the two points to graph the function.
Q: How do I graph a quadratic function?
A: To graph a quadratic function, you can follow these steps:
- Find the x-intercepts and y-intercepts of the function.
- Plot the x-intercepts and y-intercepts on a coordinate plane.
- Draw a parabola through the two points to graph the function.
Q: What is the difference between a function and a relation?
A: A function is a relation in which each input value corresponds to exactly one output value. A relation, on the other hand, is a set of ordered pairs that may or may not have a one-to-one correspondence between the input and output values.
Q: How do I determine if a relation is a function?
A: To determine if a relation is a function, you can follow these steps:
- Check if each input value corresponds to exactly one output value.
- If each input value corresponds to exactly one output value, then the relation is a function.
Q: What is the domain of a function?
A: The domain of a function is the set of all input values for which the function is defined.
Q: What is the range of a function?
A: The range of a function is the set of all output values for which the function is defined.
Q: How do I find the domain and range of a function?
A: To find the domain and range of a function, you can follow these steps:
- Check the function for any restrictions on the input values.
- Determine the set of all input values for which the function is defined.
- Determine the set of all output values for which the function is defined.
Conclusion
In conclusion, algebraic equations are a fundamental concept in mathematics, and understanding how to solve them is essential for success in many fields. By following the order of operations (PEMDAS) and using the quadratic formula, you can solve linear and quadratic equations. Additionally, understanding the difference between a function and a relation, as well as the domain and range of a function, is crucial for graphing and analyzing functions.