Solving For 'n' When X = 3y In A Linear System

by Admin 47 views
Solving for 'n' when x = 3y in a Linear System

Hey everyone! Let's dive into a fun little problem where we need to find the value of n in a system of linear equations, given that x is three times y. This type of problem combines algebra with a bit of logical thinking, so let’s break it down step by step.

Understanding the Problem

We are given the following system of equations:

  1. 4x + 3y = n + 15
  2. 3x - 5y = n - 7

Our goal is to find the value of n such that x = 3y. This condition is crucial because it allows us to reduce the number of variables and solve for n. Essentially, we'll substitute x with 3y in both equations, which will give us two equations with y and n as variables. We can then solve this new system to find the value of n.

It's like a mini puzzle where we use the relationship between x and y to unlock the value of n. So, grab your algebraic tools, and let’s get started!

Step-by-Step Solution

Step 1: Substitute x with 3y

In the first equation, 4x + 3y = n + 15, replace x with 3y:

4(3y) + 3y = n + 15

Simplify this to:

12y + 3y = n + 15

Which further simplifies to:

15y = n + 15

Now, let's do the same for the second equation, 3x - 5y = n - 7. Replace x with 3y:

3(3y) - 5y = n - 7

Simplify this to:

9y - 5y = n - 7

Which further simplifies to:

4y = n - 7

Step 2: Solve the System of Equations

Now we have a new system of equations:

  1. 15y = n + 15
  2. 4y = n - 7

We can solve this system using several methods, such as substitution or elimination. Let's use the substitution method. From the second equation, we can express n in terms of y:

n = 4y + 7

Now substitute this expression for n into the first equation:

15y = (4y + 7) + 15

Simplify and solve for y:

15y = 4y + 22

11y = 22

y = 2

Step 3: Find the Value of n

Now that we have the value of y, we can find the value of n using the expression we found earlier:

n = 4y + 7

Substitute y = 2:

n = 4(2) + 7

n = 8 + 7

n = 15

So, the value of n that satisfies the condition x = 3y in the given system of equations is 15.

Verification

To make sure our answer is correct, let’s plug n = 15 back into the original equations and find the values of x and y:

  1. 4x + 3y = 15 + 15 => 4x + 3y = 30
  2. 3x - 5y = 15 - 7 => 3x - 5y = 8

We also know that x = 3y. Substitute x with 3y in both equations:

  1. 4(3y) + 3y = 30 => 12y + 3y = 30 => 15y = 30 => y = 2
  2. 3(3y) - 5y = 8 => 9y - 5y = 8 => 4y = 8 => y = 2

Since y = 2, then x = 3y = 3(2) = 6. Now, let's verify these values in the original equations:

  1. 4x + 3y = 4(6) + 3(2) = 24 + 6 = 30 = n + 15 => n = 15
  2. 3x - 5y = 3(6) - 5(2) = 18 - 10 = 8 = n - 7 => n = 15

The values x = 6, y = 2, and n = 15 satisfy both equations and the condition x = 3y. Therefore, our solution is correct.

Alternative Methods

Elimination Method

Instead of substitution, we could have used the elimination method to solve the system of equations:

  1. 15y = n + 15
  2. 4y = n - 7

Multiply the first equation by 4 and the second equation by 15 to eliminate y:

  1. 60y = 4n + 60
  2. 60y = 15n - 105

Now set the two equations equal to each other:

4n + 60 = 15n - 105

11n = 165

n = 15

This method confirms our previous result.

Matrix Method

For those familiar with matrices, we can represent the original system of equations in matrix form and solve for n. However, this method is more complex and not as straightforward as substitution or elimination for this particular problem.

Common Mistakes to Avoid

  1. Incorrect Substitution: Make sure to correctly substitute x = 3y in both equations. A small mistake here can lead to an incorrect value for n.
  2. Algebraic Errors: Double-check your algebra when simplifying and solving the equations. Simple arithmetic errors can throw off the entire solution.
  3. Forgetting to Verify: Always verify your solution by plugging the values of x, y, and n back into the original equations to ensure they hold true.
  4. Not Simplifying: Simplify the equations as much as possible before solving. This makes the calculations easier and reduces the chance of errors.

Real-World Applications

While this problem might seem purely theoretical, solving systems of linear equations has many real-world applications. For example:

  1. Engineering: Engineers use systems of equations to design structures, analyze circuits, and optimize processes.
  2. Economics: Economists use systems of equations to model supply and demand, analyze market trends, and predict economic outcomes.
  3. Computer Science: Computer scientists use systems of equations in algorithms, data analysis, and machine learning.
  4. Physics: Physicists use systems of equations to describe motion, forces, and energy in physical systems.

Understanding how to solve these types of problems is a valuable skill that can be applied in various fields.

Conclusion

We’ve successfully found the value of n such that x = 3y in the given system of linear equations. By substituting x with 3y, we reduced the problem to a simpler system that we could easily solve. Remember to always verify your solution to ensure accuracy.

Keep practicing these types of problems to sharpen your algebra skills. Solving systems of equations is a fundamental concept in mathematics with wide-ranging applications. You guys did great! Keep up the excellent work, and you’ll become algebraic masters in no time! If you have any questions, feel free to ask. Happy solving!