how to solve 3

Question

1. Given: 2x-2y-Z=3 4x+5y-2Z=-3 3x+4y-3Z=-7 Solve for the value of Z by using Cramers rule. Determine the value of the cofactor of -3. 2. Solve for x and y if: (3-4j)^2 = [(-x-yj)/(j^2)] 3. Given: a+4b+8c=0 2a-5b=-2+6c B-4c=-3 Solve for c by the use of Cramers rules 4. Given: 3y+2x = Z+1 3x+2Z = 8 - 5y 3Z -1 = x - 2Z Determine the value of y by using Cramers rule only. 5. Solve for z by using determinants if: x+y=10 x-y=2

Answer 1

1)

Given equations:

2x  - 2y  - z  = 3 ---> (1)

4x  + 5y  - 2z  = -3 ---> (2)

3x  + 4y  - 3z  = 7 ---> (3)

Write the equations into matrices form AX  = B .

A  is coefficient matrix , X  is variable matrix and B  is constant matrix.

Now solve the equations in Cramer's method

          = - 14 - 12 -1

          = - 27

          = 94 + 74 +3

          =171

Find the value of z .

z = Det A₃ / Det A

   = 171 / -27

   = - 6.33

The value of z is - 6.33
 

 

Answer 2

2)

(3-4j)² = [(-x-yj)/(j²)]

Recall the formula :  (a - b) ² = a² - 2ab + b² and j² = -1.

3² - 2(3)(4j) + (4j)² = [(-x-yj)/(-1)]

9 - 24 j + 16 j² = x + y j

9 - 24 j - 16 = x + y j

-7 - 24j = x + y j

Compare j - coefficients and constants

x = -7

y = -24 

 

Answer 3

3)

 

Given equations:

a  + 4b  + 8c  = 0 ---> (1)

2a  - 5b  - 6c  = -2 ---> (2)

b  - 4c  = -3          ---> (3)

Write the equations into matrices form AX  = B .

A  is coefficient matrix , X  is variable matrix and B  is constant matrix.

Now solve the equations in Cramer's method

               = 1 (20 + 6) - (-8 - 0) +8 (2 - 0 )

               =26 + 8 +16

              = 50          

               = 1 (15 + 2) - (-6 - 0) +0

               =17 + 6

               = 23 

Find the value of c .

z = Det A₃ / Det A

   = 23 /50

   = 0.46

The value of c is 0.46 

Answer 4

4)

Given equations:

2x  + 3y  - z  = 1 ---> (1)

3x + 5y  + 2z  = 8 ---> (2)

x  - 5z  = -1          ---> (3)

Write the equations into matrices form AX  = B .

A  is coefficient matrix , X  is variable matrix and B  is constant matrix.

Now solve the equations in Cramer's method

               = 2 (-25 + 0) - 3 (-15 - 2) - 1 (0 - 5 )

               = -50 + 51 + 5

              = 6

 

               = 2 (-40 + 2) - 1 (-15 - 2) -1( -3 - 8)

               = -76 + 17 + 11

               = -48

Find the value of y .

y = Det A₂ / Det A

   = -48 /6

   = -8

The value of  y is -8

Answer 5

(5).

Substitution Method :

The system of equations are x + y = 10 and x - y = 2.

Step 1 : Solve the equation 1 : x + y = 10 for x since the coefficient is 1.

Subtract y from each side.

x + y - y = 10 - y
x = 10 - y

Step 2 : Substitute 10 - y for x in the equation 2 : x - y = 2 to find the value of y.

(10 - y) - y = 2

10 - 2y = 2

Subtract 10 from each side.

10 - 2y - 10 = 2 - 10

- 2y = - 8

Divide each side by negative 2.

- 2y/(-2) = - 8/(-2)

y = 4.

Step 3 : Substitute 4 for y in either equation to find x.

Equation 2: x - y = 2

x - (4) = 2

x = 6.

The solution (x, y) = (6, 4).