Given The Equation, Solve For \[$ N \$\].$\[ A - N = A - 1 + (n - 1) D \\]
Introduction
In mathematics, solving for a variable in an equation is a fundamental concept that is used to find the value of the variable. In this article, we will focus on solving for the variable n in a linear equation. We will use the given equation, a - n = a - 1 + (n - 1)d, to demonstrate the steps involved in solving for n.
Understanding the Equation
The given equation is a linear equation, which means it is an equation in which the highest power of the variable is 1. The equation is:
a - n = a - 1 + (n - 1)d
To solve for n, we need to isolate the variable n on one side of the equation.
Step 1: Simplify the Equation
The first step in solving for n is to simplify the equation by combining like terms. We can start by combining the terms on the right-hand side of the equation.
a - n = a - 1 + nd - d
Next, we can combine the constant terms on the right-hand side of the equation.
a - n = a - 1 + nd - d a - n = a - 1 + nd - d a - n = a - 1 + (n - 1)d
Step 2: Isolate the Variable n
Now that we have simplified the equation, we can isolate the variable n by moving all the terms involving n to one side of the equation. We can start by subtracting a from both sides of the equation.
-a + n = -1 + (n - 1)d
Next, we can add 1 to both sides of the equation.
n - a + 1 = (n - 1)d
Step 3: Factor Out the Common Term
The next step is to factor out the common term from the left-hand side of the equation. We can start by factoring out the term n.
n(1 - 1) - a + 1 = (n - 1)d
Since (1 - 1) = 0, we can simplify the left-hand side of the equation.
-a + 1 = (n - 1)d
Step 4: Solve for n
Now that we have isolated the variable n, we can solve for n by dividing both sides of the equation by (n - 1).
Q: What is the given equation?
A: The given equation is a - n = a - 1 + (n - 1)d.
Q: What is the goal of solving for n?
A: The goal of solving for n is to isolate the variable n on one side of the equation and find its value.
Q: What are the steps involved in solving for n?
A: The steps involved in solving for n are:
- Simplify the equation by combining like terms.
- Isolate the variable n by moving all the terms involving n to one side of the equation.
- Factor out the common term from the left-hand side of the equation.
- Solve for n by dividing both sides of the equation by (n - 1).
Q: What is the importance of simplifying the equation?
A: Simplifying the equation is important because it helps to reduce the complexity of the equation and makes it easier to isolate the variable n.
Q: What is the significance of isolating the variable n?
A: Isolating the variable n is significant because it allows us to find the value of n, which is the goal of solving for n.
Q: What is the role of factoring out the common term?
A: Factoring out the common term helps to simplify the equation further and makes it easier to solve for n.
Q: What is the final step in solving for n?
A: The final step in solving for n is to divide both sides of the equation by (n - 1) to find the value of n.
Q: What are some common mistakes to avoid when solving for n?
A: Some common mistakes to avoid when solving for n include:
- Not simplifying the equation properly
- Not isolating the variable n correctly
- Not factoring out the common term correctly
- Not dividing both sides of the equation by (n - 1) correctly
Q: How can I practice solving for n?
A: You can practice solving for n by working through examples and exercises that involve solving linear equations. You can also try solving for n in different types of equations, such as quadratic equations or polynomial equations.
Q: What are some real-world applications of solving for n?
A: Solving for n has many real-world applications, including:
- Physics: Solving for n is used to find the value of a variable in a physical equation, such as the equation of motion.
- Engineering: Solving for n is used to find the value of a variable in an engineering equation, such as the equation of a circuit.
- Economics: Solving for n is used to find the value of a variable in an economic equation, such as the equation of supply and demand.
Q: Can I use technology to solve for n?
A: Yes, you can use technology to solve for n. Many calculators and computer software programs have built-in functions that can solve for n in a linear equation.
Q: What are some tips for solving for n?
A: Some tips for solving for n include:
- Read the equation carefully and understand what is being asked.
- Simplify the equation as much as possible.
- Isolate the variable n by moving all the terms involving n to one side of the equation.
- Factor out the common term from the left-hand side of the equation.
- Solve for n by dividing both sides of the equation by (n - 1).
- Check your work by plugging the value of n back into the original equation.