Solving Systems Of Equations Using Determinants Cramer's Rule

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Hey everyone! Today, let's dive into the fascinating world of solving systems of linear equations using determinants. It might sound intimidating at first, but trust me, it's a powerful technique that can make your math life a whole lot easier. We'll walk through a specific example step-by-step, so you'll be a pro in no time!

The Problem: A System of Three Equations

So, systems of equations, what are we even talking about? Imagine you have a set of equations, like this one:

-3x + 5y - 4z = -5
4x - 2y + 3z = 2
7x + y + 5z = 7

This is a system of three linear equations with three unknowns (x, y, and z). Our goal is to find the values of x, y, and z that satisfy all three equations simultaneously. There are several ways to tackle this, but we're going to focus on the determinant method, also known as Cramer's Rule. This method provides a systematic way to solve for the variables using determinants of matrices formed from the coefficients of the equations.

Now, before we jump into the solution, let's understand why this is important. Systems of equations pop up everywhere in real life – from engineering and physics to economics and computer science. They help us model and solve problems involving multiple related variables. Mastering the techniques to solve these systems opens doors to a wide range of applications. Think about balancing chemical equations, optimizing resource allocation, or even designing bridges! The ability to solve systems of equations is a fundamental skill in many STEM fields, and the determinant method is a valuable tool in your mathematical arsenal. So, let's get to it and unlock the power of determinants!

Understanding Determinants: The Key to the Solution

Before we can use determinants to solve the system, we need to understand what a determinant is. Think of it as a special number that can be calculated from a square matrix (a matrix with the same number of rows and columns). This number holds a lot of information about the matrix, including whether the system of equations has a unique solution. Determinants are a fundamental concept in linear algebra, providing a concise way to represent certain properties of a matrix and the linear transformation it represents. The determinant gives us insight into the matrix's invertibility and the volume scaling factor of the transformation.

For a 2x2 matrix, the determinant is calculated quite simply. If we have a matrix like this:

| a  b |
| c  d |

The determinant is calculated as (ad) - (bc). It's just a simple cross-multiplication and subtraction. The determinant of a 2x2 matrix is a straightforward calculation, but it lays the groundwork for understanding determinants of larger matrices.

Things get a little more interesting with 3x3 matrices, which is what we'll be dealing with in our system of equations. There are a couple of ways to calculate the determinant of a 3x3 matrix, but one common method involves expanding along a row or column. Let's say we have a 3x3 matrix:

| a  b  c |
| d  e  f |
| g  h  i |

To find the determinant, we can expand along the first row (we could choose any row or column, but the first row is often the easiest to visualize). The formula looks like this:

determinant = a * (determinant of the 2x2 matrix formed by removing a's row and column)

  • b * (determinant of the 2x2 matrix formed by removing b's row and column)
  • c * (determinant of the 2x2 matrix formed by removing c's row and column)

Which translates to:

determinant = a * (ei - fh) - b * (di - fg) + c * (dh - eg)

Don't worry, it looks more complicated than it is! We're essentially breaking down the 3x3 determinant into a sum of 2x2 determinants, which we already know how to calculate. Understanding the calculation of a 3x3 determinant is crucial for solving systems of three linear equations. The process of expanding along a row or column, while seemingly intricate, becomes more intuitive with practice. Remember, the sign alternates between positive and negative as you move along the row or column, a key detail to keep in mind for accurate calculations.

Setting Up the Matrices: Transforming Equations into Arrays

Okay, we've got the determinant concept down. Now, how do we apply it to our system of equations? The first step is to represent our system as matrices. Matrices are simply rectangular arrays of numbers, and they're perfect for organizing the coefficients in our equations. Setting up matrices correctly is crucial for the accurate application of Cramer's Rule. A clear understanding of how the coefficients and constants translate into matrix form is essential to avoid errors in the subsequent calculations.

We'll create a few matrices. First, we have the coefficient matrix (let's call it matrix A), which consists of the coefficients of our variables (x, y, and z):

A = |
 -3  5  -4 |
 |  4 -2   3 |
 |  7  1   5 |

Notice how we just take the numbers in front of x, y, and z and arrange them in a grid. The first row corresponds to the first equation, the second row to the second equation, and so on. The columns represent the coefficients of x, y, and z, respectively. The coefficient matrix is the backbone of our system, holding the key numerical relationships between the variables.

Next, we need to create matrices where we replace one column of matrix A with the constants on the right-hand side of our equations. These constants form a column vector:

B = |
 -5 |
 |  2 |
 |  7 |

We'll use this column vector to create three new matrices:

  • Matrix Ax: Replace the first column of A (the x-coefficient column) with B.
  • Matrix Ay: Replace the second column of A (the y-coefficient column) with B.
  • Matrix Az: Replace the third column of A (the z-coefficient column) with B.

So, these matrices will look like this:

Ax = |
 -5  5  -4 |
 |  2 -2   3 |
 |  7  1   5 |

Ay = |
 -3 -5  -4 |
 |  4  2   3 |
 |  7  7   5 |

Az = |
 -3  5  -5 |
 |  4 -2   2 |
 |  7  1   7 |

We now have four matrices: A, Ax, Ay, and Az. These matrices hold all the information we need to solve for x, y, and z using Cramer's Rule. The careful construction of these matrices is a crucial step in the determinant method. Each matrix represents a specific manipulation of the original system, allowing us to isolate and solve for each variable individually. So, double-check your matrices to ensure accuracy before moving on to the determinant calculations. A small error in the matrix setup can lead to significant errors in the final solution. Let's calculate the determinant of each of these matrices to proceed with the solution.

Calculating Determinants: The Nitty-Gritty

Alright, the matrices are set, now for the fun part – calculating those determinants! This is where the formula we discussed earlier comes into play. We'll be calculating the determinants of a 3x3 matrix, so remember the expansion method. Calculating determinants is the heart of Cramer's Rule, and while it might seem tedious, it's a systematic process that ensures we arrive at the correct solution. Accuracy is key here, as even a small mistake in the determinant calculation can throw off the entire result. So, let's take our time, double-check our work, and master this crucial skill.

Let's start with the determinant of matrix A (denoted as det(A)). We'll expand along the first row:

det(A) = -3 * ((-2)5 - 31) - 5 * (45 - 37) + (-4) * (4*1 - (-2)*7)

Let's break it down step-by-step:

  • -3 * ((-10) - 3) = -3 * (-13) = 39
  • -5 * (20 - 21) = -5 * (-1) = 5
  • -4 * (4 - (-14)) = -4 * 18 = -72

So, det(A) = 39 + 5 - 72 = -28

Now, let's calculate det(Ax). Again, expanding along the first row:

det(Ax) = -5 * ((-2)5 - 31) - 5 * (25 - 37) + (-4) * (2*1 - (-2)*7)

  • -5 * ((-10) - 3) = -5 * (-13) = 65
  • -5 * (10 - 21) = -5 * (-11) = 55
  • -4 * (2 - (-14)) = -4 * 16 = -64

So, det(Ax) = 65 + 55 - 64 = 56

Next up, det(Ay):

det(Ay) = -3 * (25 - 37) - (-5) * (45 - 37) + (-4) * (47 - 27)

  • -3 * (10 - 21) = -3 * (-11) = 33
  • 5 * (20 - 21) = 5 * (-1) = -5
  • -4 * (28 - 14) = -4 * 14 = -56

So, det(Ay) = 33 - 5 - 56 = -28

Finally, det(Az):

det(Az) = -3 * ((-2)7 - 21) - 5 * (47 - 27) + (-5) * (4*1 - (-2)*7)

  • -3 * ((-14) - 2) = -3 * (-16) = 48
  • -5 * (28 - 14) = -5 * 14 = -70
  • -5 * (4 - (-14)) = -5 * 18 = -90

So, det(Az) = 48 - 70 - 90 = -112

Phew! We've calculated all the determinants. This was the most computationally intensive part, but we made it through. Now, we're ready to use these determinants to find the values of x, y, and z. These determinant values are the building blocks for applying Cramer's Rule. Each determinant represents a scaled version of the solution space, and their ratios will give us the values of our variables. So, let's move on to the final step and see how these numbers translate into the solution of our system of equations.

Cramer's Rule: Unveiling the Solution

Okay, we've reached the grand finale! We've calculated all the necessary determinants, and now we're ready to use Cramer's Rule to find the values of x, y, and z. Cramer's Rule provides a direct formula for solving for each variable using the determinants we just calculated. Cramer's Rule is the elegant culmination of all our previous work, providing a clear and concise formula to extract the solution from the calculated determinants. It's a powerful tool that transforms complex calculations into a straightforward method for finding the values of the variables.

Cramer's Rule states the following:

x = det(Ax) / det(A) y = det(Ay) / det(A) z = det(Az) / det(A)

See? It's quite simple. We just divide the determinant of the matrix where we replaced the corresponding column with the constant vector by the determinant of the original coefficient matrix.

Let's plug in the values we calculated:

x = 56 / (-28) = -2 y = -28 / (-28) = 1 z = -112 / (-28) = 4

And there you have it! We've found the solution to the system of equations:

x = -2, y = 1, z = 4

This means that the point (-2, 1, 4) is the solution that satisfies all three equations simultaneously. The solution we've obtained is the unique intersection point of the three planes represented by the equations. This geometrical interpretation provides a visual understanding of what we've accomplished through the algebraic process of solving the system.

To be absolutely sure, it's always a good idea to check our solution by plugging these values back into the original equations:

  • -3*(-2) + 51 - 44 = 6 + 5 - 16 = -5 (Correct!)
  • 4*(-2) - 21 + 34 = -8 - 2 + 12 = 2 (Correct!)
  • 7*(-2) + 1 + 5*4 = -14 + 1 + 20 = 7 (Correct!)

Our solution checks out! We've successfully solved the system of equations using determinants and Cramer's Rule. This verification step is crucial in ensuring the accuracy of our solution. By substituting the obtained values back into the original equations, we confirm that they indeed satisfy all the conditions of the system.

Conclusion: Determinants for the Win!

Guys, we did it! We successfully solved a system of three equations using the determinant method. It might have seemed like a lot of steps at first, but hopefully, you now see how this powerful technique works. Determinants provide a systematic and elegant way to solve linear systems, and mastering this method will definitely boost your math skills.

Remember, practice makes perfect. The more you work with determinants and Cramer's Rule, the more comfortable you'll become with the process. Try solving different systems of equations, and you'll start to recognize patterns and develop a deeper understanding of linear algebra. This method is not just a mathematical trick; it's a fundamental tool with applications in various fields. So, embrace the power of determinants and keep exploring the fascinating world of mathematics!