Use Cramer's Rule to solve (if possible) the system of equations. { 4 x-2 y+3 z=-2 2 x

Use Cramer's Rule to solve (if possible) the system of...

Key Concepts

Determinant

The determinant is a scalar value derived from a square matrix that offers important information about the matrix properties, such as invertibility. In the context of linear systems, a nonzero determinant of the coefficient matrix indicates that the system has a unique solution, while a zero determinant suggests that the system either has no solution or infinitely many solutions.

Cramer’s Rule

Cramer’s Rule is a mathematical theorem used to solve systems of linear equations with as many equations as unknowns, provided that the determinant of the coefficient matrix is nonzero. It involves calculating determinants of modified matrices where one column is replaced with the constant terms, allowing each variable’s value to be found as a ratio of determinants.

System of Linear Equations

A system of linear equations is a collection of equations where each equation is linear in the unknowns. The goal when working with such systems is to find values for the variables that satisfy all equations simultaneously. These systems can have a unique solution, infinitely many solutions, or no solution depending on the relationships between the equations.

Coefficient Matrix

The coefficient matrix is formed by the coefficients of the variables in a system of linear equations. It encapsulates the structure of the linear system and is a key element in many solution techniques, including Cramer's Rule, since its determinant provides crucial information about the system’s solvability.

Transcript

00:01 According to the question, we have to solve the system of equations using krammer's rule.

00:05 So the equations are given by 4x minus 2y plus 3 z equals minus of 2.

00:14 Let this be our first equation.

00:16 Then the second equation is given by 2x plus 2y plus 5 z equals 16.

00:27 Let this be our second equation and the final equation is given by.

00:33 8x minus of 5y minus of 2 z equals 4 so this is our third equation now at first in order to solve this question using grammar's rule we have to find the value of the coefficient matrix so the coefficient matrix is determined by del and it is a matrix which has been formed using the coefficients of all the terms so for example the first column or so the first row of this coefficient matrix is formed by the coefficients of the first equation so 4 minus 2 and 3 now similarly the second row of the coefficient matrix is formed by the coefficients of the second equation so it will be 2 2 and 5 and finally the third column is formed by the coefficients of the 3 equation so it is 8 minus of 5 and minus of 2 this upon further simplification will give us minus of 82 so the value of determinant is coefficient matrix is minus of 82 and since it is not equal 0 so we can solve it further now the value of x can be given by del x upon del the value of y is given by del and the value of z is given by del z upon del where del x del x del x del where del x del x del is are three unique determinants now what is del x? dahl x is a determinant in which the coefficients of y and z are remain as it is but in place of writing the coefficients of x we are going to replace it by the constant term so in the case of the first equation the coefficient of x is 4 so instead of writing 4 we are going to write the constant term that is minus of 2 so here we write minus of 2 similarly for the second equation instead of writing 2 where we will write 16 and for the third equation, instead of writing 8, we will write 4.

02:36 And for the coefficients of y and z, 8 will remain as it is.

02:40 So we will have minus 2, 2 and minus 5.

02:44 And here we will write the coefficients of z, so 3, 5, and minus of 2.

02:51 This upon further simplification will give us, we will give del x equals minus of 410...

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