Solving Linear Equations Using Determinants A Step-by-Step Guide

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Hey guys! Let's dive into a cool math problem where we're going to use determinants to solve for the variables x and y in a system of linear equations. It might sound a bit intimidating at first, but trust me, it's a super useful tool once you get the hang of it. We'll break it down step by step, making sure everyone can follow along. So, grab your pencils, and let’s get started!

The System of Equations

First off, let's take a look at the system of linear equations we're dealing with:

4x + 16y = -68
12x + 3y = 21

Our mission, should we choose to accept it (and we do!), is to figure out the values of x and y that make both of these equations true at the same time. There are a few ways we could tackle this, like substitution or elimination, but today we're going to focus on using determinants. Determinants offer a systematic way to solve these kinds of problems, and they're especially handy when you're working with larger systems of equations.

What are Determinants?

Okay, so what exactly is a determinant? In simple terms, a determinant is a special number that can be calculated from a square matrix (a matrix with the same number of rows and columns). For a 2x2 matrix, which is what we'll be using today, the determinant is calculated like this:

For a matrix:
| a  b |
| c  d |

The determinant is: ad - bc

Basically, you multiply the numbers on the main diagonal (from top-left to bottom-right) and subtract the product of the numbers on the other diagonal (from top-right to bottom-left). Easy peasy, right? This might seem a little abstract now, but you'll see how it fits into solving our system of equations in just a moment.

Setting Up the Matrices

To use determinants, we first need to rewrite our system of equations in matrix form. This involves creating a few different matrices:

  • Coefficient Matrix (A): This matrix is formed by the coefficients of our variables (x and y). In our case, it looks like this:

    | 4  16 |
| 12  3 |

    See how the 4 and 12 are the coefficients of x, and the 16 and 3 are the coefficients of y? They're just arranged in the same order as they appear in the equations.

  • Variable Matrix (X): This is a simple column matrix containing our variables:

    | x |
| y |

  • Constant Matrix (B): This matrix contains the constants on the right side of our equations:

    | -68 |
| 21 |

Cracking the Code with Cramer's Rule

Now for the magic! We're going to use something called Cramer's Rule to solve for x and y. Cramer's Rule is a formula that uses determinants to find the solutions to a system of linear equations. Here's how it works:

  • To find x:

x = det(A_x) / det(A) ```

Where:

*   det(A_x) is the determinant of a matrix formed by replacing the first column of the coefficient matrix (A) with the constant matrix (B).
*   det(A) is the determinant of the original coefficient matrix (A).

  • To find y:

y = det(A_y) / det(A) ```

Where:

*   det(A_y) is the determinant of a matrix formed by replacing the second column of the coefficient matrix (A) with the constant matrix (B).
*   det(A) is the determinant of the original coefficient matrix (A).

So, we're calculating a couple of new determinants and dividing them by the determinant of our original coefficient matrix. Let's put this into action!

Calculating the Determinants

Alright, time to roll up our sleeves and do some calculations. First, we need to find the determinant of the coefficient matrix (A):

A = | 4  16 |
    | 12  3 |

det(A) = (4 * 3) - (16 * 12) = 12 - 192 = -180

So, det(A) is -180. Now, let's find det(A_x). Remember, we replace the first column of A with the constant matrix B:

A_x = | -68  16 |
      | 21   3 |

det(A_x) = (-68 * 3) - (16 * 21) = -204 - 336 = -540

Next up, det(A_y). This time, we replace the second column of A with the constant matrix B:

A_y = | 4  -68 |
      | 12  21 |

det(A_y) = (4 * 21) - (-68 * 12) = 84 + 816 = 900

We've got all the determinants we need! Now we can plug them into Cramer's Rule to solve for x and y.

Solving for x and y

Let's start with x:

x = det(A_x) / det(A) = -540 / -180 = 3

So, x is 3. Now for y:

y = det(A_y) / det(A) = 900 / -180 = -5

And there you have it! y is -5.

Putting It All Together

By using determinants and Cramer's Rule, we've successfully solved the system of equations. We found that x = 3 and y = -5. You can always double-check your answer by plugging these values back into the original equations to make sure they hold true.

Determinants might seem a little complex at first, but they're a powerful tool for solving systems of linear equations, especially when things get more complicated. Keep practicing, and you'll be a determinant pro in no time!

Identifying the Correct Determinant Setup

Now, let’s circle back to the original question and pinpoint the correct determinant setup for solving this system. Remember, the core of using determinants (specifically Cramer's Rule) is setting up the matrices correctly. We need to calculate three determinants: det(A), det(A_x), and det(A_y).

det(A): This is the determinant of the coefficient matrix, which we've already established. It's the determinant of the matrix formed by the coefficients of x and y in our equations.

det(A_x): To find this, we replace the x-coefficient column in matrix A with the constants from the right-hand side of the equations.

det(A_y): Similarly, for this determinant, we replace the y-coefficient column in matrix A with the constants.

Why This Works: A Deeper Dive

Cramer's Rule isn't just a magic formula; it has a solid mathematical foundation. It's based on the properties of determinants and how they relate to the area (in 2D) or volume (in 3D) spanned by the vectors represented by the matrix columns. When we replace a column with the constant vector, we're essentially finding the scaled area or volume in a particular direction, which then allows us to isolate the value of one variable.

For those who are interested in the theory behind the method, here's a more in-depth look:

The system of equations can be represented in matrix form as:

AX = B

Where:

  • A is the coefficient matrix.
  • X is the variable matrix.
  • B is the constant matrix.

If det(A) is not zero, we can find the inverse of A (A⁻¹). Multiplying both sides of the equation by A⁻¹ gives us:

A⁻¹AX = A⁻¹B

Since A⁻¹A is the identity matrix (I), we have:

IX = A⁻¹B
X = A⁻¹B

Cramer's Rule provides a way to calculate the elements of X (the variables) without explicitly calculating A⁻¹. It uses determinants to achieve this, making it an elegant and efficient method.

Common Pitfalls and How to Avoid Them

  • Incorrect Matrix Setup: The most common mistake is setting up the matrices incorrectly. Double-check that you've placed the coefficients and constants in the right columns and rows.
  • Forgetting the Order of Operations: Remember the determinant formula (ad - bc) and the order in which you multiply and subtract. A simple sign error can throw off the entire calculation.
  • Dividing by Zero: If det(A) is zero, Cramer's Rule cannot be used directly. This indicates that the system of equations either has no solution or infinitely many solutions. In such cases, other methods like Gaussian elimination or substitution might be more appropriate.

Real-World Applications of Linear Equations and Determinants

Linear equations and determinants aren't just abstract math concepts; they have tons of real-world applications. Here are a few examples:

  • Engineering: Solving systems of equations is crucial in structural analysis, circuit design, and fluid dynamics.
  • Economics: Linear models are used to analyze supply and demand, market equilibrium, and economic growth.
  • Computer Graphics: Matrices and determinants are fundamental in 3D graphics for transformations like rotations, scaling, and projections.
  • Cryptography: Linear algebra plays a role in encryption and decryption algorithms.
  • Data Analysis: Solving systems of equations can help in regression analysis and other statistical modeling techniques.

Understanding how to solve linear equations is a foundational skill in many fields, and determinants provide a powerful tool for tackling these problems.

Conclusion

So, there you have it! We've journeyed through the world of determinants, learned how to use Cramer's Rule, and solved for x and y in our system of linear equations. Remember, the key is to break down the problem into smaller steps: setting up the matrices, calculating the determinants, and applying Cramer's Rule. With a little practice, you'll be solving systems of equations like a pro!

I hope this guide has been helpful and has demystified determinants for you. Keep exploring the fascinating world of mathematics, and you'll be amazed at the power and beauty of these tools.