# Cramer's Rule and Determinants

• Deagonx
In summary, to solve the linear system using Cramer's rule, you first need to create a matrix using the constants accompanying the variables. Then, find the determinant of this matrix by multiplying diagonally down from the top left and subtracting that from the product of multiplying diagonally up from the bottom left. Next, create a new matrix by replacing the first column with the constants from the equal values, leaving the second column unchanged. Finally, calculate the determinant of the new matrix and divide it by the determinant of the original matrix to find the solution for y. To find the solution for x, substitute the value of y into one of the equations and solve for x, or use the same substitution method with matrices.
Deagonx

## Homework Statement

Use Cramer's rule to solve the linear system.

## Homework Equations

(only showing one, I think if one is explained I will figure out the rest)
2x - y = -2
x + 2y = 14

What I'm told I'm supposed to do, is to take the constants accompanying the variables and make a matrix out of them (2, -1, 1, 2)
Then find the determinant. To find the determinant, multiply diagonally down from the top left, and subtract that from the product of multiplying diagonally up from the bottom left. If I'm doing this right, I get 4 - (-1) which would give me 5. I found the determinant, and I have no idea what to do from there.

## The Attempt at a Solution

What I think I'm supposed to do going off a rough memory, is take a matrix from the equaled values (-2 and 14) and put them on the right of the matrix, then use the Xs on the left. Find the determinant of that, and divide it by 5 (the determinant of the first one.) Of course, I really have no idea.
So if I do what I think I'm supposed to do (which I'm quite sure is the wrong thing) I get 28 - (-2) and get 30. 30 divided by 5 is 6. I think 6 is y.

So if y is 6, then 2x - 6 = -2, so 2x = 4. x = 2. So x is 2, x + 2y is 2 + 2(6) which is actually 14.

So maybe I got it right, but I'd like a bit of confirmation.

Last edited:
You have the correct idea for the y coefficients. Instead of plugging in y into one of the equations and solving for x, you can just do the same substitution into the first column (leaving the second alone) and calculating its determinant.

If you're still confused, I can illustrate it with matrices, but everything you did was correct!

## 1. What is Cramer's Rule?

Cramer's Rule is a mathematical formula used to solve systems of linear equations. It involves using determinants to find the solution to the equations.

## 2. How is Cramer's Rule used to solve systems of equations?

Cramer's Rule involves finding the determinants of the coefficient matrix and the constant matrix, and then using these determinants to find the values of the variables in the system of equations.

## 3. What are determinants?

Determinants are mathematical values associated with a square matrix. They can be used to solve systems of equations, find the inverse of a matrix, and determine linear independence.

## 4. Are there any limitations to using Cramer's Rule?

Yes, Cramer's Rule can only be used to solve systems of equations with the same number of equations as variables. It also becomes increasingly complex and time-consuming as the number of variables increases.

## 5. How does Cramer's Rule relate to other methods of solving systems of equations?

Cramer's Rule is just one method of solving systems of equations. It can be used in conjunction with other methods, such as substitution or elimination, to check the accuracy of the solution or to solve systems with more than one variable.

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