Solve the system for z.

2x - y + 2z = 11
x - 2y - 3z = -5
-3x + z = -3

Solve by linear equation x+y=1,3y+3z=+3x=4

x=4

y=-1
z=3

To solve the system for z, we need to use substitution or elimination method. Let's use the elimination method in this case.

Step 1: Look for a variable that has the same coefficient in two equations. In this case, the third equation has a coefficient of 1 for z.

Step 2: Multiply the third equation by the necessary factors to make the coefficient of z equal to the coefficient in either the first or second equation. Since the coefficient of z in the third equation is already 1, we don't need to multiply it.

-3x + z = -3 (equation 3)
Multiply equation 3 by 2 to make the coefficient of z equal to 2 in the first equation:
-6x + 2z = -6 (equation 4)

Step 3: Add the two equations that have the same coefficient for z to eliminate z.

2x - y + 2z = 11 (equation 1)
-6x + 2z = -6 (equation 4)
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-4x - y = 5 (equation 5)

Step 4: Now we have eliminated the variable z. We can substitute this value back into one of the original equations to solve for the remaining variables.

We'll substitute equation 5 into equation 2:
x - 2y - 3z = -5 (equation 2)
x - 2y - 3(-3) = -5
x - 2y + 9 = -5
x - 2y = -14 (equation 6)

Step 5: Now we have two equations with two variables:

-4x - y = 5 (equation 5)
x - 2y = -14 (equation 6)

We can solve this system using the elimination method:

Multiply equation 6 by 4 to make the coefficient of x -4:
4x - 8y = -56 (equation 7)

Add equation 7 to equation 5 to eliminate x:
-4x - y + 4x - 8y = 5 - 56
-9y = -51

Divide both sides by -9 to solve for y:
y = -51 / -9
y = 17/3 or y ≈ 5.67

Step 6: Now that we have the value of y, we can substitute it back into equation 6 to solve for x:

x - 2(17/3) = -14
x - 34/3 = -14
x = -14 + 34/3
x = (-42 + 34) / 3
x = -8/3 or x ≈ -2.67

Step 7: Finally, substitute the values of x and y back into equation 3 to solve for z:

-3x + z = -3
-3(-8/3) + z = -3
8 + z = -3
z = -3 - 8
z = -11

Therefore, the solution for the system of equations is x = -8/3, y = 17/3, and z = -11.

Matrix mathematics capter

-x+3y-z+w=-2

X-y+3z-w=0
y-z+4w=7
4x-y+z=5

- 3 x + z = - 3 Add 3 x to both sides

- 3 x + 3 x + z = - 3 + 3 x

z = 3 x - 3

x - 2 y - 3 z = - 5

x - 2 y - 3 * ( 3 x - 3
) = - 5

x - 2 y - 3 * 3 x - 3 * ( - 3 = - 5

x - 2 y - 9 x + 9 = - 5

- 8 x - 2 y + 9 = - 5 Add 8 x - 9 to both sides

- 8 x - 2 y + 9 + 8 x - 9 = - 5 + 8 x - 9

- 2 y = - 14 + 8 x Divide both sides by - 2

y = - 14 / - 2 + 8 x / - 2

y = 7 - 4 x

2 x - y + 2 z = 11

2 x - ( 7 - 4 x ) + 2 * ( 3 x - 3 ) = 11

2 x - 7 + 4 x + 6 x - 6 = 11

12 x - 13 = 11 Add 13 to both sides

12 x - 13 + 13 = 11 + 13

12 x = 24 Divide both sides by 12

x = 24 / 12

x = 2

y = 7 - 4 x

y = 7 - 4 * 2

y = 7 - 8

y = - 1

z = 3 x - 3

z = 3 * 2 - 3

z = 6 - 3

z = 3

Solution :

x = 2 , y = - 1 , z = 3