For Exercises 11–22, use Cramer's Rule to solve each system.

Question

For Exercises 11–22, use Cramer's Rule to solve each system. $\begin{cases}2x = 3y + 2 \5x = 51 - 4y\end{cases}$

Accepted Answer

Hey everyone welcome back in this problem. We are asked to solve the following system of equations were told to use Cramer's rule. Okay. And the system of equations were given is eight plus three. Y equals 11 and eight X plus seven Y is equal to 54. Okay. Alright, so getting started in order to use Cramer's rule. We want to write this system in terms of matrices and vectors. Okay, So we want to write this in the form of A. X equals B. Okay, we're a. S and matrix we can do this by doing the following. Okay. So we're gonna take the coefficient relating to X. The first equation, the coefficient relating to Y in the first equation. Okay, that's gonna be the first row of our matrix and then the same in the second row relating to the second equation, the coefficient corresponding to the X. And the corresponding or sorry, the coefficient corresponding to the Y. Okay, so we get the matrix with the Rose 1387. Okay. We're gonna multiply by a vector X. And the vector X is going to contain our variable. So in this case our variables are X and Y. So this is going to be the vector X. Y. Okay. And B. Is this vector of the right hand side as we get 54. Okay. And we can see that this does equal those that system were given. Okay. If we were to do the matrix vector multiplication because we get one times X plus three times Y. Alright, so that means this matrix is a our vector X. Is X. Y. And R vector B is 11 and 54. Alright, so now let's go ahead and use Cramer's rule, Cramer's rule tells us that the value of X. Or the first thing in our X vector. So in this case it's X. Okay, is going to equal to the determinant of A one divided by the determinant of A. You may be asking what is a one? A one is going to be the matrix A. Where the first column is replaced by B. Okay, so let me write that down. Okay, so we're taking the determinant of the matrix a whose first column is replaced with B. Okay, so the first column is going to be 11 And then the second column is gonna be the normal A. 37. We're gonna divide that by the determinant of a. Which is one 387. Okay, now we're just doing the determinate of two by two matrices. Okay. And so we can multiply on the diagonal and subtract On the numerator. We get 11 times -54 times three. And then the denominator we get one times -8 times three. This gives us negative 85 over negative 17 which gives an X. Value of five. Okay. Alright, so we have X value of five here. Now we do the same for why? In this case why is the second thing in our X vector? Okay, so instead of being the determinant of A two or sorry, a one it's going to be the determinant of a to get ahead of myself there. Okay, so this comes from the position in that vector. Okay, so determining a two divided by the determinant of A. Okay, what is A? Too? Well, that's the matrix where the second column has been replaced by B. Okay, so now the first column is gonna be the normal column of a 18 and the second column is going to be be 11 50 for the denominator is going to be the same. The determinant of a. The determinant of column 118 column 237. Alright, working this out, we know the denominator is negative 17 because that's the same determinant that we found for X. In the numerator we have one times 54 -8 times 11 which gives -34/-17 which is equal to two. Okay, so this is our Y value and so our solution Is going to be that X is equal to five and Y is equal to two. That's going to correspond with answer B. That's it for this one. Thanks everyone for watching. See you in the next video

For Exercises 11–22, use Cramer's Rule to solve each system. $\begin{cases}\)2x = 3y + 2 \\5x = 51 - 4y\(\end{cases}$

Key Concepts

Systems of Linear Equations

Cramer's Rule

Determinants of 2x2 Matrices

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