Welcome to a video on Cramer's Rule. This is a way of solving a system of equations using determinants. Let's first take a look at a system of two equations with two unknowns.
Notice the a's are the coefficient of the x terms, the b's are the coefficient of the y terms, and the c's are the constants. So if the system is set up this way, x is equal to this quotient and y is equal to this quotient. If we first take a look at the denominator of these quotients, The elements in the determinant are formed by the coefficients of the x and y terms. If we take a look at the numerator for the value of x, if we replace the x coefficients with the constants, we have the numerator for the value of x. Similarly for y, if we take out the y coefficients and replace them with the constants, we have the numerator for the value of y.
Let's go and take a look at an example. Let's go ahead and see if we can solve this using determinants. We know the values of x and y will be a quotient of determinants. So let's go ahead and set up our quotients. Next, the determinant in the denominator will be the two by two determinant formed by the coefficient of the x and y terms.
So the first row will be one, negative one. And the second row will be one, four. Both for x and y. Now the numerator for the value of x will come from replacing the x coefficients with the constant terms. So we'll take out the one and the one and replace them with negative three and seventeen.
And this column stays the same. For y, we're going to replace the y coefficients with the constants. So we'll take out the negative one and four and replace it with negative three and seventeen.
and the x column stays the same. Let's go ahead and evaluate these determinants. Here we'll have negative 12 minus negative 17, that'll become negative 12 plus 17, that's five. The denominator will be four minus negative one, or four plus one, that's five.
And remember the denominators are the same, so this will also be five. And the numerator for the y value will be 17. minus negative three or seventeen plus three, that'll be twenty. So the solution to the system will be x equals one and y equals four.
Let's go and take a look at a system of three equations and three variables. Now it looks like a lot's going on here, but the pattern does remain the same for this type of system. And what I mean by that is if you look at the denominators of each of these variables, It's a determinant formed by the coefficients of the x, y, and z terms. Next, if we want to solve for the x value, the numerator will come from replacing the x coefficients with the constants.
The numerator for y will come from replacing the y coefficients with the constants. And the numerator for z will come from replacing the z coefficients with the constants. So let's go ahead and try one of these as well.
So we'll have x, y, and z. Each will be a quotient of determinants. So each of the determinants in the denominator will come from the coefficients of these three equations. So the first row will be two, three, one.
Negative one, two, three. And negative three, negative three, one. Remember each of these denominators will be the same. Now the numerator for x will come from the determinant form by replacing the x coefficients with the constants two, negative one, zero.
So we'll have two, negative one, zero. And the next two columns will stay the same. The numerator for y will come from replacing the y coefficients with the constants two, negative one, zero.
Column one and column three will stay the same. And for the numerator of z, we'll replace the z coefficients with two, negative one, and zero. And the first two columns stay the same.
Let's go ahead and evaluate these on the graphing calculator. So we'll press second, matrix, go over to edit, press enter. Let's go ahead and calculate the denominators of all of these first. Let's go back to the home screen. Let's find the value of these denominators.
So we'll find the determinant of matrix A. Second matrix, math, enter. Second matrix, A. Close that. And it's equal to seven.
So each of these denominators is equal to seven. Now let's go ahead and evaluate each of the numerators. Let's go ahead and put the numerators in as matrix B, C, and D. So two, three, one. Negative one, two, three.
Zero, negative three, one. Let's go ahead and enter this in as matrix C, three by three. So we have two, two, one. Negative one, negative one, three.
And negative three, zero, one. And then let's go to enter in matrix D. Let's go ahead and move the calculator for that one.
First row is two, three, two. Negative one, two, negative one. Negative three, negative three, zero.
Now let's go ahead and determine the numerators of these values. So remember the numerator for x would be the determinant of matrix B. So second matrix math, enter matrix B.
So second matrix B. We have twenty-eight. Let's go ahead and find the determinant of matrix C.
Second matrix math, enter. Second matrix, that would be matrix C. That's negative 21. And then for the numerator of Z, we'll have the determinant of, which is equal to 21. So we'll have 28, negative 21, and 21. These all come out very nice.
X is equal to four, y is equal to negative three, and z is equal to positive three. So Cramer's rule would be quite a bit of work to do by hand, but if you're allowed to use technology, it's a great way to use what we've learned about matrices and determinants to solve a system of equations. Thank you for watching.