# Linear Algebra Examples

, ,

Step 1

Step 1.1

Move .

Step 1.2

Subtract from both sides of the equation.

Step 1.3

Subtract from both sides of the equation.

Step 1.4

Move all terms containing variables to the left side of the equation.

Step 1.4.1

Subtract from both sides of the equation.

Step 1.4.2

Add to both sides of the equation.

Step 1.5

Move .

Step 1.6

Reorder and .

Step 2

Represent the system of equations in matrix format.

Step 3

Step 3.1

Write in determinant notation.

Step 3.2

Choose the row or column with the most elements. If there are no elements choose any row or column. Multiply every element in row by its cofactor and add.

Step 3.2.1

Consider the corresponding sign chart.

Step 3.2.2

The cofactor is the minor with the sign changed if the indices match a position on the sign chart.

Step 3.2.3

The minor for is the determinant with row and column deleted.

Step 3.2.4

Multiply element by its cofactor.

Step 3.2.5

The minor for is the determinant with row and column deleted.

Step 3.2.6

Multiply element by its cofactor.

Step 3.2.7

The minor for is the determinant with row and column deleted.

Step 3.2.8

Multiply element by its cofactor.

Step 3.2.9

Add the terms together.

Step 3.3

Multiply by .

Step 3.4

Evaluate .

Step 3.4.1

The determinant of a matrix can be found using the formula .

Step 3.4.2

Simplify the determinant.

Step 3.4.2.1

Simplify each term.

Step 3.4.2.1.1

Multiply by .

Step 3.4.2.1.2

Multiply .

Step 3.4.2.1.2.1

Multiply by .

Step 3.4.2.1.2.2

Multiply by .

Step 3.4.2.2

Add and .

Step 3.5

Evaluate .

Step 3.5.1

The determinant of a matrix can be found using the formula .

Step 3.5.2

Simplify the determinant.

Step 3.5.2.1

Simplify each term.

Step 3.5.2.1.1

Multiply by .

Step 3.5.2.1.2

Multiply by .

Step 3.5.2.2

Subtract from .

Step 3.6

Simplify the determinant.

Step 3.6.1

Simplify each term.

Step 3.6.1.1

Multiply by .

Step 3.6.1.2

Multiply by .

Step 3.6.2

Add and .

Step 3.6.3

Add and .

Step 4

Since the determinant is not , the system can be solved using Cramer's Rule.

Step 5

Step 5.1

Replace column of the coefficient matrix that corresponds to the -coefficients of the system with .

Step 5.2

Find the determinant.

Step 5.2.1

Choose the row or column with the most elements. If there are no elements choose any row or column. Multiply every element in row by its cofactor and add.

Step 5.2.1.1

Consider the corresponding sign chart.

Step 5.2.1.2

The cofactor is the minor with the sign changed if the indices match a position on the sign chart.

Step 5.2.1.3

The minor for is the determinant with row and column deleted.

Step 5.2.1.4

Multiply element by its cofactor.

Step 5.2.1.5

The minor for is the determinant with row and column deleted.

Step 5.2.1.6

Multiply element by its cofactor.

Step 5.2.1.7

The minor for is the determinant with row and column deleted.

Step 5.2.1.8

Multiply element by its cofactor.

Step 5.2.1.9

Add the terms together.

Step 5.2.2

Multiply by .

Step 5.2.3

Evaluate .

Step 5.2.3.1

The determinant of a matrix can be found using the formula .

Step 5.2.3.2

Simplify the determinant.

Step 5.2.3.2.1

Simplify each term.

Step 5.2.3.2.1.1

Multiply by .

Step 5.2.3.2.1.2

Multiply .

Step 5.2.3.2.1.2.1

Multiply by .

Step 5.2.3.2.1.2.2

Multiply by .

Step 5.2.3.2.2

Add and .

Step 5.2.4

Evaluate .

Step 5.2.4.1

The determinant of a matrix can be found using the formula .

Step 5.2.4.2

Simplify the determinant.

Step 5.2.4.2.1

Simplify each term.

Step 5.2.4.2.1.1

Multiply by .

Step 5.2.4.2.1.2

Multiply by .

Step 5.2.4.2.2

Subtract from .

Step 5.2.5

Simplify the determinant.

Step 5.2.5.1

Simplify each term.

Step 5.2.5.1.1

Multiply by .

Step 5.2.5.1.2

Multiply by .

Step 5.2.5.2

Subtract from .

Step 5.2.5.3

Add and .

Step 5.3

Use the formula to solve for .

Step 5.4

Substitute for and for in the formula.

Step 6

Step 6.1

Replace column of the coefficient matrix that corresponds to the -coefficients of the system with .

Step 6.2

Find the determinant.

Step 6.2.1

Choose the row or column with the most elements. If there are no elements choose any row or column. Multiply every element in row by its cofactor and add.

Step 6.2.1.1

Consider the corresponding sign chart.

Step 6.2.1.2

The cofactor is the minor with the sign changed if the indices match a position on the sign chart.

Step 6.2.1.3

The minor for is the determinant with row and column deleted.

Step 6.2.1.4

Multiply element by its cofactor.

Step 6.2.1.5

The minor for is the determinant with row and column deleted.

Step 6.2.1.6

Multiply element by its cofactor.

Step 6.2.1.7

The minor for is the determinant with row and column deleted.

Step 6.2.1.8

Multiply element by its cofactor.

Step 6.2.1.9

Add the terms together.

Step 6.2.2

Multiply by .

Step 6.2.3

Evaluate .

Step 6.2.3.1

The determinant of a matrix can be found using the formula .

Step 6.2.3.2

Simplify the determinant.

Step 6.2.3.2.1

Simplify each term.

Step 6.2.3.2.1.1

Multiply by .

Step 6.2.3.2.1.2

Multiply by .

Step 6.2.3.2.2

Subtract from .

Step 6.2.4

Evaluate .

Step 6.2.4.1

The determinant of a matrix can be found using the formula .

Step 6.2.4.2

Simplify the determinant.

Step 6.2.4.2.1

Simplify each term.

Step 6.2.4.2.1.1

Multiply by .

Step 6.2.4.2.1.2

Multiply by .

Step 6.2.4.2.2

Subtract from .

Step 6.2.5

Simplify the determinant.

Step 6.2.5.1

Simplify each term.

Step 6.2.5.1.1

Multiply by .

Step 6.2.5.1.2

Multiply by .

Step 6.2.5.2

Add and .

Step 6.2.5.3

Add and .

Step 6.3

Use the formula to solve for .

Step 6.4

Substitute for and for in the formula.

Step 7

Step 7.1

Replace column of the coefficient matrix that corresponds to the -coefficients of the system with .

Step 7.2

Find the determinant.

Step 7.2.1

Step 7.2.1.1

Consider the corresponding sign chart.

Step 7.2.1.2

The cofactor is the minor with the sign changed if the indices match a position on the sign chart.

Step 7.2.1.3

The minor for is the determinant with row and column deleted.

Step 7.2.1.4

Multiply element by its cofactor.

Step 7.2.1.5

The minor for is the determinant with row and column deleted.

Step 7.2.1.6

Multiply element by its cofactor.

Step 7.2.1.7

The minor for is the determinant with row and column deleted.

Step 7.2.1.8

Multiply element by its cofactor.

Step 7.2.1.9

Add the terms together.

Step 7.2.2

Evaluate .

Step 7.2.2.1

The determinant of a matrix can be found using the formula .

Step 7.2.2.2

Simplify the determinant.

Step 7.2.2.2.1

Simplify each term.

Step 7.2.2.2.1.1

Multiply by .

Step 7.2.2.2.1.2

Multiply .

Step 7.2.2.2.1.2.1

Multiply by .

Step 7.2.2.2.1.2.2

Multiply by .

Step 7.2.2.2.2

Subtract from .

Step 7.2.3

Evaluate .

Step 7.2.3.1

The determinant of a matrix can be found using the formula .

Step 7.2.3.2

Simplify the determinant.

Step 7.2.3.2.1

Simplify each term.

Step 7.2.3.2.1.1

Multiply by .

Step 7.2.3.2.1.2

Multiply by .

Step 7.2.3.2.2

Add and .

Step 7.2.4

Evaluate .

Step 7.2.4.1

The determinant of a matrix can be found using the formula .

Step 7.2.4.2

Simplify the determinant.

Step 7.2.4.2.1

Simplify each term.

Step 7.2.4.2.1.1

Multiply by .

Step 7.2.4.2.1.2

Multiply by .

Step 7.2.4.2.2

Add and .

Step 7.2.5

Simplify the determinant.

Step 7.2.5.1

Simplify each term.

Step 7.2.5.1.1

Multiply by .

Step 7.2.5.1.2

Multiply by .

Step 7.2.5.1.3

Multiply by .

Step 7.2.5.2

Subtract from .

Step 7.2.5.3

Add and .

Step 7.3

Use the formula to solve for .

Step 7.4

Substitute for and for in the formula.

Step 7.5

Divide by .

Step 8

List the solution to the system of equations.