In Problems 15-42, solve each system of equations using...

In Problems $15-42,$ solve each system of equations using Cramer's Rule if it is applicable. If Cramer's Rule is not applicable, say so.
$$\left\{\begin{array}{l}
x+2 y=5 \\
x-y=3
\end{array}\right.$$

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Key Concepts

Cramer's Rule

A method of solving a system of linear equations by using determinants. It applies to square systems where the coefficient matrix has a non-zero determinant, allowing the solution for each variable to be found by replacing the corresponding column with the constant terms and calculating the ratio of determinants.

Systems of Linear Equations

A collection of two or more linear equations involving the same set of variables. These systems can have a unique solution, infinitely many solutions, or no solution, and they are a fundamental aspect of linear algebra and algebra. The methods for solving them include graphing, substitution, elimination, and using determinants among others.

Determinant

A scalar value computed from the elements of a square matrix which provides useful properties, such as determining the invertibility of a matrix. In the context of Cramer’s Rule, a non-zero determinant of the coefficient matrix indicates that the system has a unique solution.

Transcript

0:00 Hello.

00:01 So here we have the system of equations.

00:03 We have x plus 2 y is equal to 5 and x minus y is equal to 3.

00:07 Okay, so we're going to first find the determinant d of the coefficient matrix.

00:13 So the coefficient matrix is just going to be, well, 1, 2, 1 negative 1.

00:21 Also, finding the determinant here is just equal to, again, the product of the main diagonal.

00:27 So that's 1 times a negative 1 and then minus the product of this diagonal.

00:33 So minus 2 times 1.

00:35 So that gives us a minus 1 minus 2, which is equal to negative 3.

00:41 Right.

00:41 And because first we go ahead, we find the coefficient of the coefficient matrix and then we get something, if we get any value that's non -zero, we can use kramer's rule.

00:52 So because we had that d here is not equal to 0, right, negative 3, not equal to, zero, we can then use kramer's rule.

01:01 Okay.

01:03 So we're going to find now d sub x and d sub y.

01:07 So d sub x, well, that is found by replacing the first column with the solution, 5 -3.

01:18 So we'll replace the first column with 5 -3, and what we get is 5 -3 in the first column, and then we have 2 -9 -1.

01:26 Then we find the determinant of this 2 by 2...

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