solve each system of equations using Cramer’s Rule if it is applicable. If Cramer’s Rule is not

Solve each system of equations using Cramer’s Rule if it is...

Key Concepts

Cramer's Rule

Cramer's Rule is a method for solving systems of linear equations with an equal number of equations and unknowns. It involves replacing a column of the coefficient matrix with the constant terms and finding the determinant of the new matrix. Each variable is then solved by dividing this determinant by the determinant of the original coefficient matrix, provided that the latter is not zero.

System of Linear Equations

A system of linear equations is a set of equations with multiple variables that are solved together. The objective is to find values for each variable that satisfy every equation in the system simultaneously. In this context, it involves two equations with two unknowns.

Coefficient Matrix

The coefficient matrix is a matrix composed solely of the coefficients from the system of linear equations. It represents the linear relationships between the variables and is used in determining the feasibility of some solution methods, including Cramer's Rule.

Determinant

The determinant is a specific scalar value computed from a square matrix. For systems of linear equations using Cramer's Rule, the determinant of the coefficient matrix is crucial; if this determinant is non-zero, the system has a unique solution, and Cramer's Rule is applicable.

Transcript

00:01 In this problem, we are giving a system of equations, and our goal is to solve using kramer's rule if applicable.

00:08 So in order to do this, we first must find the determinant of the 2x2 matrix that will be built out of the coefficients of x and y from the left side of the equation.

00:22 So in order to do that, we must recall that if we have a 2x2 matrix, maybe it's made up of a, b, c, and d, we can find the determinant by focusing on the diagonals.

00:32 The blue diagonal results in a positive and the red diagonal results in a negative.

00:37 So we have the final determinant to be a times d minus b times c.

00:42 So we're going to first build that determinant out of the coefficients of x and y.

00:50 So the x's will go in the first column on the left, the 2 and the 10.

00:54 And then i take the negative 3 and the positive 10 from the y coefficients for the right -hand column.

00:59 And now i focus by multiplying on the diagonals, blue is the 3.

01:03 Positive, red is negative.

01:05 So the determinant of the coefficients is 2 times 10 to give us 20, minus negative 3 times 10 to give us negative 30.

01:13 A double negative, as you know, turns into a positive.

01:17 So now we have determinant is 20 plus 30 to give us 50.

01:21 And i'm going to circle that to hang on to it for our final step, but first we have to move into step two.

01:29 If the determinant had come out to be zero, kramer's rule could not be applied.

01:43 And this would be because either the system is inconsistent or because there's infinitely many solutions.

01:49 However, we did not get there.

01:51 We found the determinant to be not zero.

01:54 In our case, it was 50.

01:56 So now we calculate the determinant of x and the determinant of y by building two more two by two matrices.

02:08 So the first one for the x's focuses on replacing the coefficients of x with the right -hand side of our original equation.

02:17 So those coefficients of x get replaced by negative 1 and 5, while the right side keeps those coefficients of y...

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