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.
$$\left\{\begin{aligned}
4 x+5 y &=-3 \\
-2 y &=-4
\end{aligned}\right.$$

Key Concepts

System of Linear Equations

A system of linear equations is a collection of two or more equations that involve the same set of variables. The goal is to find values for these variables that satisfy all the equations simultaneously. Methods to solve such systems include substitution, elimination, and matrix techniques such as Cramer's Rule.

Applicability of Cramer's Rule

Cramer's Rule can only be applied to square systems of linear equations, where the number of equations is equal to the number of unknowns. Moreover, for the rule to yield a unique solution, the determinant of the coefficient matrix must be non-zero; if it is zero, the system does not have a unique solution or is not suitable for this method.

Cramer's Rule

Cramer's Rule is a method for solving systems of linear equations using determinants. It states that if the coefficient matrix of a square system has a non-zero determinant, then each variable can be solved as the ratio of two determinants: one for the coefficient matrix and one where a column is replaced with the constants from the equations.

Determinant

The determinant is a scalar value derived from a square matrix that indicates whether the matrix is invertible. In the context of solving linear equations, a non-zero determinant confirms the existence of a unique solution, while a zero determinant indicates that the system may be either dependent or inconsistent.

Transcript

0:00 Hello.

00:01 So here is our given system of equations.

00:03 So first, we're going to find the determinant d of the coefficient matrix.

00:09 So therefore we have that d is equal to the two by two determinant where we have our coefficients in our first equation are four and five.

00:19 So we have four and five.

00:22 And then in the second equation, we have no x.

00:24 So the coefficient on x is well zero and then we have negative two so zero negative two so the two by two determine here is just four times negative two which is negative eight and then minus five times zero so minus zero which is equal to negative eight so therefore right d is equal to negative eight which is not equal to zero therefore yes we can use kramer's rule okay so kramer's rule is nice where it just says that um we just find d sub x and d sub y where d sub y where d x is found by just replacing the first column by the constants on the right hand side of the equation.

01:06 So by the column vector negative 3, negative 4.

01:10 So d sub x is going to be equal to the 2 by 2 determinants, where now we have negative 3, negative 4 as our first column, and then we have 5 negative 2.

01:20 So we have d sub x is equal to negative 3 times negative 2, which is a positive 6, and then minus 5 ,000.

01:28 Times negative 4.

01:29 So that's minus a, well, minus a negative 20, which is plus 20, right, which is equal to 6 plus 20, which is 26.

01:38 So d sub x is 26.

01:40 And d sub y, d sub y is equal to, again, the 2 by 2 determinant.

01:45 Now we replace the second column with the column vector, negative 3, negative 4.

01:50 So we have d sub y, first column, 4, second column, negative 3, negative 4...

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