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. $\begin{cases}3x - 4y = 4 \2x + 2y = 12\end{cases}$

Accepted Answer

Hey everyone, welcome back in this problem. We are asked to solve the following system of equations. Using Cramer's rule. The equations were given our four X minus Y equals 23 X plus three. Y. Is equal to 36. Now we're using Cramer's rule. The first thing we want to do is write these equations or the system of equations into matrix vector form. Okay, so we want to write these equations in the form. A X is equal to B. Okay, we're a is a two by two matrix. Alright, so we can do that by doing the following. We take the coefficient corresponding to the X term in the first column. Mhm. And the coefficient corresponding to the second, sorry to the Y term in the second column. We keep the first equation in the first row and the second equation in the second row. Okay, this gives us a matrix A first row. Four negative one and second row. 33. Okay, our vector X is going to be the vector of variables. So we have X. Y. Okay. And our vector B is going to be the vector of the constants on the right hand side. So in this case 23, And you can think about multiplying this matrix vector product out to double check. So we have we would get four times X minus one times Y. Okay. And same on the on the second room. Alright, now that we have our matrix vector product we just need to go ahead and use Cramer's rule. What is Cramer's rule. Tell us. Well it tells us that the value of X can be given by the determinant of a one divided by the determinative. A. What is a one? A One is the matrix that is made by removing the first column of A. And replacing it with B. Okay, so we're gonna get the determinant of the matrix where we replace column one with B. This is going to give us the matrix with first column 26. And second column negative 13. Okay. We're going to divide that by the determinant of a negative 13. All right now we're just calculating the two x 2 determinants. So we just multiply the diagonals. Okay recall we multiply the diagnose and subtract. So in the numerator we're going to get 23 times 3 -26 times negative one. Okay and in the denominator four times three minus three times negative one. Okay and just watch your signs. It can be easy to mix up signs here. This gives us 105 over 15 which is equal to seven. Okay, so now we've found an X. Value of seven. Okay, we need to find out why value in order to have a full solution. Okay, so for why we're going to do the same thing. We're gonna have the determinant. But in this case instead of a one we have a two. Okay. And that too comes because it's the second thing in the X vector. Okay. It's got index two in the X. Factor. So because why is here we get this A two divided by a determinant of a. Alright, so a two similarly to a. One. We're going to replace the second column of A. With B. Okay. So we're going to take the determinant of the matrix whose first column is 43 and his second column is 23. 26. Okay. We're going to divide it by the determinative. A Determined 4. 3 -1. 3. Alright. On the numerator we get four times 26 -3 times 23 In the denominator we know that this determinant is equal to 15. Okay. Until we get 75 divided by 15 which is equal to five. Okay So we have our X. Value and our Y. Value. And we can write, our solution is X. Is equal to seven and Y is equal to five. Okay, so this is our solution here that's going to correspond with answer D. That's it for this one. Thanks everyone for watching. I hope this video helped

For Exercises 11–22, use Cramer's Rule to solve each system. $\begin{cases}\)3x - 4y = 4 \\2x + 2y = 12\(\end{cases}$

Key Concepts

Cramer's Rule

Determinants of 2x2 Matrices

Systems of Linear Equations

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