Question

5х+y-3z= -2 4x+3y+2z= 16 2x-3y+z= 17

Answer 1

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Answer 2

To solve the system of equations:

1. 5x + y - 3z = -2
2. 4x + 3y + 2z = 16
3. 2x - 3y + z = 17

There are several methods to solve this system, such as substitution, elimination, or matrix methods. Let's use the elimination method to find the values of x, y, and z.

Step 1: Eliminate z from equations 1 and 2.

To do this, we need to make the coefficients of z in both equations the same, but with opposite signs. Let's multiply equation 1 by 2 and equation 2 by 3:

2(5x + y - 3z) = 2(-2) => 10x + 2y - 6z = -4 3(4x + 3y + 2z) = 3(16) => 12x + 9y + 6z = 48

Now, add the two equations together:

(10x + 2y - 6z) + (12x + 9y + 6z) = -4 + 48 22x + 11y = 44

Step 2: Eliminate z from equations 1 and 3.

Similarly, let's multiply equation 1 by 1 and equation 3 by 3:

1(5x + y - 3z) = 1(-2) => 5x + y - 3z = -2 3(2x - 3y + z) = 3(17) => 6x - 9y + 3z = 51

Now, add the two equations together:

(5x + y - 3z) + (6x - 9y + 3z) = -2 + 51 11x - 8y = 49

Step 3: Now, we have two equations:

1. 22x + 11y = 44
2. 11x - 8y = 49

We can solve these two equations simultaneously. Let's multiply the second equation by 2 and then add the two equations together to eliminate y:

2(11x - 8y) = 2(49) => 22x - 16y = 98

(22x + 11y) + (22x - 16y) = 44 + 98 44x - 5y = 142

Step 4: Now, we have two equations:

1. 44x - 5y = 142
2. 11x - 8y = 49

Again, we can solve these two equations simultaneously. Let's multiply the second equation by 4 and then subtract the two equations to eliminate x:

4(11x - 8y) = 4(49) => 44x - 32y = 196

(44x - 5y) - (44x - 32y) = 142 - 196 -27y = -54

Step 5: Solve for y:

y = -54 / -27 y = 2

Step 6: Now that we have the value of y, we can substitute it back into equation 2 to find x:

11x - 8y = 49 11x - 8(2) = 49 11x - 16 = 49 11x = 49 + 16 11x = 65 x = 65 / 11 x = 5

Step 7: Now that we have the values of x and y, we can substitute them back into equation 1 to find z:

5x + y - 3z = -2 5(5) + 2 - 3z = -2 25 + 2 - 3z = -2 27 - 3z = -2 -3z = -2 - 27 -3z = -29 z = -29 / -3 z = 29/3

So, the solution to the system of equations is x = 5, y = 2, and z = 29/3 (approximately 9.67).