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

Accepted Answer

Hey everyone, welcome back in this problem. We are asked to solve the following system of equations using Cramer's rule. Okay, the equations were given our two X plus nine? Y equals 11 and five X minus 12. Y is equal to 16. Okay, first thing we want to do when we're using Cramer's rule is right. Our system of equations in matrix vector form. Okay, so we want to write it in the form A X is equal to B. Where A is a two by two matrix and X and B are vectors. Alright, we can do that by doing the following a is going to be the matrix of coefficients. Okay, so we're gonna take the coefficients corresponding with the first equation in the top row. Okay, so corresponding with X. We have to in the first column and corresponding with nine or sorry, with Y. We have nine in the second column. In the same for equation to in the second row we get five and negative 12. Our x vector is going to be our variable. So in this case we get x, Y and r. B vector is going to be those values on the right hand side of the equation. 11 and 16. Okay. And so you can imagine if you multiplied out this matrix times this vector, you're going to get two times X plus nine times Y equals 11. In the top five, X minus 12, Y equals 16 in the bottom. Okay, so it's the exact same system. Just rewritten using matrices and vectors. Okay, now we want to apply Cramer's rule. Cramer's rule tells us that The value of X will be given by the determinant of a one divided by the determinant of a. Okay, what is a one? A One is going to be the matrix A. Where the first column is replaced by B. Okay. And we have the subscript one here because X is the first thing in this X vector. Okay. Alright, so the matrix where the first column is replaced by B. So we're gonna have column one is going to be 11 16 and the second column is going to be nine negative 12. Okay, like normal matrix A. And then the determinant of a on the denominator um along the first row to nine and along the second row, five negative 12. These are determinants of two by two matrices case or recall we multiply on the diagonals and subtract. We're going to have 11 times negative 12 minus 16 times nine in the numerator and the denominator, two times negative 12 minus five times nine. Okay, and just pay attention to your signs. It can be easy to mix them up here. So we're going to get negative 276 in the numerator over negative 69. And the denominator for an X value of four. Okay, so we found our X value using Cramer's rule. Now we do the exact same thing for Y. Okay, so Y is equal to the determinant and in this case instead of having a one we get a two. Okay we get a two because why is the second thing in the X. Factor? Okay we use the index of the expect er in this case is to we divide it by the determinant of a gun. Well this is going to give us A two. Is going to be a where the second column is replaced by B. So we get the determinant of in the first column to five and in the second column 11 16 The determinant of a we found above which is -69. OK, so we leave that we'll save ourselves doing that work again in the numerator we get two times 16 minus five times 11 divided by negative 69 and we get negative 23 divided by negative 69 which gives a Y. Value of one third. So we found our X. Value and we found our Y value. So we have the solution Is X. is equal to four and Y is equal to 1/3. That's the solution that solves this system. Okay and that's going to correspond with answer. C. Thanks everyone for watching. I hope this video helped see you in the next one

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

Key Concepts

Cramer's Rule

Determinants of 2x2 Matrices

Solving Systems of Linear Equations

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