Question

Solve the following system of equations for all three variables

5x+7y+4z=10
9x+4y+4z=-9
9x+2y-z=6

X=? y=? z=?

Answer 1

To solve this system of equations, we can use the method of elimination or substitution. Here, we will use substitution.

1) Solve the first equation for x:
5x + 7y + 4z = 10
5x = 10 - 7y - 4z
x = 2 - (7/5)y - (4/5)z

2) Substitute the expression for x in the second and third equations:
9(2 - (7/5)y - (4/5)z) + 4y + 4z = -9
9(2 - (7/5)y - (4/5)z) + 2y - z = 6

3) Simplify the equations:
18 - 9(7/5)y - 9(4/5)z + 4y + 4z = -9
18 - 9(7/5)y - 9(4/5)z + 2y - z = 6

4) Further simplify the equations:
18 - (63/5)y - (36/5)z + 4y + 4z = -9
18 - (63/5)y - (36/5)z + 2y - z = 6

5) Combine like terms:
18 + (4y - (63/5)y) + (4z - (36/5)z) = -9
18 + ((8/5)y - (36/5)z) + (2y - z) = 6

6) Simplify further and combine like terms:
18 + (-(43/5)y - (4/5)z) = -9
18 + ((18/5)y - (1/5)z) = 6

7) Subtract 18 from both sides:
-(43/5)y - (4/5)z = -27
(18/5)y - (1/5)z = -12

Now, we have a system of two equations in two variables. We can solve it using the same method. Let's solve it using elimination:

1) Multiply the first equation by 5 and multiply the second equation by 5 to eliminate fractions:
-43y - 4z = -135
18y - z = -60

2) Multiply the second equation by 4 and add it to the first equation to eliminate the z variable:
-43y - 4z + 72y - 4z = -135 + (-240)
29y - 8z = -375

3) Simplify the equation:
29y - 8z = -375

We now have a system of two equations with two variables:
-43y - 4z = -135
29y - 8z = -375

We can solve this system using the method of elimination or substitution. Here, we will use elimination:

1) Multiply the first equation by 29 and multiply the second equation by -43:
-1257y - 116z = -3915
-1257y + 344z = 16125

2) Add the two equations to eliminate the y variable:
-1257y -116z + (-1257y + 344z) = -3915 + 16125
-116z + 344z = 12210
228z = 12210
z = 12210 / 228
z = 53.55

3) Substitute the value of z back into one of the original equations to find the value of y:
-43y - 4(53.55) = -135
-43y - 214.2 = -135
-43y = 214.2 - 135
-43y = 79.2
y = 79.2 / -43
y = -1.84

4) Substitute the values of y and z back into the expression for x:
x = 2 - (7/5)(-1.84) - (4/5)(53.55)
x = 2 + (12.88/5) - (214.2/5)
x = 2 + 2.576 - 42.84
x = -38.264

Therefore, the solution to the system of equations is:
x = -38.264, y = -1.84, z = 53.55.