Question

2x-3y-z=-22
4x+5y+2z=21
-7x+2y-3z=37

Answer 1

The given system of equations is:

2x - 3y - z = -22 ...(1)

4x + 5y + 2z = 21 ...(2)

-7x + 2y - 3z = 37 ...(3)

To find the solution, we can use a method such as Gaussian elimination or matrix operations. Here, I will use Gaussian elimination to solve the system of equations.

Step 1: Multiply equation (1) by 2 and equation (3) by 4 to eliminate the x variable:

4x - 6y - 2z = -44 ...(4)

-28x + 8y - 12z = 148 ...(5)

Step 2: Add equation (2) and equation (5) to eliminate the x variable:

4x + 5y + 2z = 21 ...(2)

-28x + 8y - 12z = 148 ...(5)

----------------------------

-23x + 13y - 10z = 169 ...(6)

Step 3: Multiply equation (4) by 7 and equation (6) by 2 to eliminate the x variable:

28x - 42y - 14z = -308 ...(7)

-46x + 26y - 20z = 338 ...(8)

Step 4: Add equation (7) and equation (8) to eliminate the x variable:

28x - 42y - 14z = -308 ...(7)

-46x + 26y - 20z = 338 ...(8)

----------------------------

-16y - 34z = 30 ...(9)

Step 5: Multiply equation (9) by 3/8 to simplify the coefficients:

(-3/2)y - (17/4)z = 15/4 ...(10)

Step 6: Multiply equation (10) by 8 to eliminate the fraction:

-12y - 34z = 30 ...(11)

Step 7: Add equation (11) and equation (6) to eliminate the y variable:

-12y - 34z = 30 ...(11)

-23x + 13y - 10z = 169 ...(6)

----------------------------

-23x - 21z = 199 ...(12)

Step 8: Multiply equation (12) by -2 to simplify the coefficients:

46x + 42z = -398 ...(13)

Step 9: Add equation (13) and equation (2) to eliminate the z variable:

46x + 42z = -398 ...(13)

4x + 5y + 2z = 21 ...(2)

----------------------------

50x + 5y = -377 ...(14)

Step 10: Multiply equation (14) by -1/5 to simplify the coefficients:

-10x - y = 75 ...(15)

Step 11: Add equation (15) and equation (6) to eliminate the x variable:

-10x - y = 75 ...(15)

-23x - 21z = 199 ...(12)

----------------------------

-24z = -124 ...(16)

Step 12: Solve equation (16) for z:

z = (-124)/(-24)

Answer 2

Explanation:

You have a system of linear equations:

2x - 3y - z = -22

4x + 5y + 2z = 21

-7x + 2y - 3z = 37

You can solve this system of equations using methods such as substitution or elimination. Here, I'll use the elimination method to solve it.

First, let's multiply equation 1 by 2, so it becomes:

4x - 6y - 2z = -44

Now, add equation 2 to this new equation:

(4x + 5y + 2z) + (4x - 6y - 2z) = 21 - 44

This simplifies to:

8x - y = -23

Now, you have a system of two equations:

8x - y = -23

-7x + 2y - 3z = 37

Let's solve equation 1 for y:

y = 8x + 23

Now, substitute this expression for y into equation 2:

-7x + 2(8x + 23) - 3z = 37

Simplify and solve for x:

-7x + 16x + 46 - 3z = 37

9x - 3z = -9

Divide the equation by 3:

3x - z = -3

Now, we have two equations:

3x - z = -3

8x - y = -23

Let's solve equation 2 for x:

8x = y - 23

Now, substitute this expression for x into equation 1:

3(y - 23) - z = -3

Expand and simplify:

3y - 69 - z = -3

Now, add z to both sides:

3y - 69 = z - 3

Now, add 69 to both sides:

3y = z - 3 + 69

3y = z + 66

Finally, divide by 3:

y = (z + 66)/3

So, the solution to your system of equations is:

x = (y - 23)/8

y = (z + 66)/3

z can take any value.

These equations represent a family of solutions since z can vary freely, and x and y will adjust accordingly based on these equations.