Question

Solve the literal equation for n. $$ \frac{3+4 n}{n}=7 b $$

Short answer

The solution to the equation is \(n = \frac{3}{7b - 4}\).

Step-by-step solution

Remove the Fraction

First, let's remove the fraction by multiplying every term by \(n\). This gives: \(3 + 4n = 7bn\).

Isolate the Terms with \(n\) on one side of the equation

Next, let's subtract \(4n\) from both sides to place all terms with \(n\) together. That results in: \(3 = 7bn - 4n.\) We can factor out \(n\) from the right side of the equation to get: \(3 = n(7b - 4)\).

Solve for \(n\)

Now, we can find \(n\) by dividing both sides by \(7b - 4\). Our \(n\) is therefore: \(n = \frac{3}{7b - 4}\).

Key concepts

Isolate Variables

Isolating variables is a core skill in algebra that helps to solve equations effectively. To isolate a variable means to get the variable by itself on one side of the equation. For example, if you have an equation like \( 3 + 4n = 7bn \), to isolate \( n \), we need to make sure that \( n \) is the only term on one side.

• First, identify all the terms containing the variable.
• Use addition or subtraction to move other terms to the opposite side.
• Finally, use division or multiplication if necessary to isolate the variable.

In our exercise, isolating \( n \) was crucial. Initially, we subtracted \( 4n \) from both sides to have all \( n \) terms on one side, simplifying our task in solving the equation.

Factoring

Factoring is a way of simplifying expressions, making it easier to find solutions to equations. It involves expressing an equation or expression as a product of its factors. In our example, after moving terms around, we factored \( n \) out of the expression \( 7bn - 4n \) which became \( n(7b - 4) \). Here's how factoring works:

• Look for common factors in all terms of an expression.
• Extract these factors out of the expression.
• Write the remaining expression as a product.

By factoring, you convert the equation into a form that is simpler to solve. Notice how reducing the original equation simplifies the path to isolating the variable and solving for \( n \).

Solving Equations

Solving equations often comes down to transforming the equation into one that is simpler to work with. Start by simplifying the equation as much as possible and isolate the desired variable.When we solved \( n \), we transformed \( 3 + 4n = 7bn \) into \( 3 = n(7b - 4) \) by:

• Eliminating fractions by multiplying every term by \( n \).
• Collecting terms to one side to make factoring easier.
• Finally, dividing by the remaining coefficient or factor to find \( n \).

This process highlights the importance of a structured approach to equation solving. By breaking down complex problems into simple, manageable steps, you can find solutions more easily.

Algebraic Manipulation

Algebraic manipulation involves rearranging equations, expressions, or formulas to simplify or solve them. This is achieved through various techniques such as:

• Adding or subtracting terms on both sides of the equation.
• Multiplying or dividing all terms by a non-zero number.
• Using distribution to simplify expressions.

In our original problem, we used algebraic manipulation to transform \( \frac{3+4n}{n} = 7b \) into \( n = \frac{3}{7b - 4} \). Each step, like removing the fraction by multiplying by \( n \) or dividing by \( 7b - 4 \), involved rearranging parts of the equation through manipulation. By mastering algebraic manipulation, you can translate complex equations into simpler forms, unlocking quicker paths to the solution.