# Solve the following equations using Cramer's Rule: 5x-4y+6z=58 -4x+6y+3z=-13 6x+3y+7z=53

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According to Cramer's Rule we can write the given set of equations in the matrix form as follows:

| 5 -4 6| | x | | 58 |

|-4 6 3| | y | = |-13 |

| 6 3 7 | | z | | 53 |

Now the value of the determinant for the matrix

| 5 -4 6|

|-4 6 3|

| 6 3 7 |

is -307, let us name this as D

x is given by

| **58** -4 6|

|**-13** 6 3|

| **53** 3 7 |

divided by D

The value of the determinant of

| **58** -4 6|

|**-13** 6 3|

| **53** 3 7 |

is -1228. Therefore x = -1228/ -307 = 4

Similarly

y is given by

| 5 **58** 6|

|-4** -13** 3|

| 6 ** 53** 7 |

divided by D

The value of the determinant of

| 5 **58** 6|

|-4** -13** 3|

| 6 ** 53** 7 |

is 614. Therefore y = 614/ -307 = -2

z is given by

| 5 -4 **58** |

|-4 6 **-13**|

| 6 3 **53** |

divided by D

The value of the determinant of

| 5 -4 **58** |

|-4 6 **-13**|

| 6 3 **53** |

is -1535. Therefore z = -1525/ -307 = 5

**Therefore x = 4, y= -2 and z= 5.**

5x-4y+6z=58..................(1)

-4x+6y+3z=-13................(2)

6x+3y+7z=53....................(3)

(1) - 2*(2)

5x - 4y +6z = 58

8x - 12y - 6z = 26

==> 13x - 16y = 84 ............(4)

-7*(2) + 3*(3).

==> 28x - 42y -21z = 91

==> 18x + 9y + 21z = 159

==> 46x - 33y = 250............(5)

==>33*(4) + (-16*(5)

==> 429x - 528y = 2772

==> -736x +528y = -4000

==> -307x = -1228

==> **x= 4**

==> 13x - 16y = 84

==> 13*4 - 16y = 84

==> 52 - 16y = 84

==> -16y = 32

**==> y= -2**

==> -4x+6y+3z=-13

==> -4*4 + 6*-2 + 3z = -13

==> -16 - 12 + 3z = -13

==> 3z = 15

**==> z = 5**

We can apply Cramer's rule if and only if the determiant of the system is different from zero.

We'll calculate the determinant:

det A = 5*6*7 + (-4)*3*6 + 3*6*(-4) - 6*6*6 - 3*3*5 - 7*16

det A = 210 - 2*4*18 - 216 - 45 - 112

det A = 210 - 517

det A = -307

Since det A is different from zero, we'll apply Cramer's rule:

x = det X/detA, y = detY/detA, z = detZ/detA

5x-4y+6z=58

-4x+6y+3z=-13

6x+3y+7z=53

detX = 58*6*7 - 13*18 - 12*53 - 53*36 - 9*58 + 13*28

detX = 2436 - 234 - 636 - 1908 - 522 + 364

det x = -1228

x = -1228/-307

**x = 4**

y = 614/-307

**y = -2**

In the same way, we'll determine det y and det z, substituting the column of the coefficients of the variable taht has to be determined, by the column of the coefficients of the right side.

Now, we'll substitute x and y in the first equation:

20 + 8 + 6z = 58

6z = 58 - 28

6z = 30

**z = 5**

**The solution of the system is {4 ; -2 ; 5}.**