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. $\begin{cases}12x + 3y = 15 \2x - 3y = 13\end{cases}$

Accepted Answer

Hey everyone, welcome back in this problem. We are asked to solve the following system of equation using Cramer's rule. Okay. System of equations were given is eight X plus 12, Y equals 12 and four x minus three, Y is equal to 15. Now when we're using Cramer's rule, what we wanna do, we want to take this system of equations and write it in matrix vector form. Okay, so we want to write it in the form, A. X equals B. Where A is a matrix and X and B are vectors. Okay? And in this case A is going to be a two by two matrix because we have two equations in two variables. Ok, so A is going to be a vector of coefficients? Okay, in the first row we have the coefficients corresponding with the first equation. In the first column we have the coefficient corresponding with the X. Okay, And then in the second row corresponding with the Y. So we get 8, 12 and in the second row we get the same thing for the second equation. So we get four negative three. The vector X is going to contain our variables X and y. The vector B is going to be the right hand side of this equation. So we have 12 and 15. Okay, so we've written it in the A. X equals B. Matrix vector form. Now let's go ahead and apply Cramer's rule. Cramer's rule tells us that the value of X can be given by the determinant of A. One divided by the determinant of a. Okay, and this one comes from X. The value we're looking for being the first thing in the X. Vector? Okay. So its index one in the X. Vector. And so we use a subscript one here when we're talking determinant of A. One. What is A one? Well when we're doing Cramer's rule A one means take the matrix A. And replace the first column with the vector. B. Okay. So we're gonna have the determinant where the first column is 15 and the second column is 12 negative three. We're gonna divide by the determinant of a first column. 84. 2nd column 12 negative three. Alright. And these should these are all determinants cannot matrices. Alright, So in the numerator we have determinant of a two by two. Matrix. Ok, let's recall. We're going to multiply the diagonals and subtract. So we get 12 times negative 3 -12 times 15. And on the denominator we get eight times negative -4 times 12. Alright, This is gonna give us negative 216 divided by negative 72. For an X value of three. This is the X value we found using Cramer's rule. Now we need to go ahead and find the corresponding Y value. So why is going to be equal to the determinant of a. And the subscript here. What's it going to be? What we're looking at. Why? Why is the second thing In the x vector. And so we're gonna get the determinant of a. two. Okay. Divided by the determinant of A now A. Two instead of replacing the first column of A. With B. We're gonna replace the second column of A. With B. Okay. So we're gonna get the determinant of the matrix with column one 84. And call them to 1215. Divided by the determinant of a. Which we know is negative 72. Okay because we found it in the denominator when we looked for X. Alright. So the numerator we get eight times -12 times four. And that's divided by 72. It gives us 72 divided by negative 72 which is equal to negative one. Okay? We have our X. Value, we have our Y. Value and so we have our solutions. X. Is equal to three. Y. Is equal to negative one. Okay. And that's the solution we found using Cramer's rule that's going to correspond with answer C. That's it for this video. I hope that helped. Thanks for watching

For Exercises 11–22, use Cramer's Rule to solve each system. $\begin{cases}12x + 3y = 15 \\2x - 3y = 13\end{cases}$

Key Concepts

Cramer's Rule

Determinants of 2x2 Matrices

Solving Systems of Linear Equations

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