Question

Solve the following system of equations for all three variables.

10x – 2y - 5z = 0
8x + 3y + 5z = 10
9x + y + 5z = -10

Answer 1

To solve the system of equations, we can use the method of elimination or substitution. Using the method of elimination, multiply and subtract equations to eliminate variables one by one until you are left with a system of equations with two variables. Solve the resulting system and substitute the values back into the original equations to find the values of the remaining variable. Therefore, the solution to the given system of equations is x = -4, y = -22, and z = 8.

Step-by-step explanation:

To solve the system of equations 10x - 2y - 5z = 0, 8x + 3y + 5z = 10, and 9x + y + 5z = -10, we can use the method of elimination or substitution.

Let's solve the system using the method of elimination:

1. Multiply the second equation by 5 to make the coefficients of z in the second and third equations equal. This gives us 40x + 15y + 25z = 50.
2. Subtract the first equation from the modified second equation to eliminate the z variable. This gives us 30x + 17y = 50.
3. Multiply the third equation by 2 to make the coefficients of z in the second and third equations equal. This gives us 18x + 2y + 10z = -20.
4. Subtract the first equation from the modified third equation to eliminate the z variable. This gives us 8x + 4y = -20.
5. Multiply the second equation by 4 to make the coefficients of y in the second and third equations equal. This gives us 32x + 12y + 20z = 40.
6. Subtract the modified third equation from the modified second equation to eliminate the y variable. This gives us 24x + 8y = 60.
7. Now we have a system of two equations with two variables: 30x + 17y = 50 and 24x + 8y = 60.
8. Multiply the second equation by 4 to make the coefficients of x in the second and third equations equal. This gives us 96x + 32y = 240.
9. Subtract the modified second equation from the first equation to eliminate the x variable. This gives us 6x + 9y = -190.
10. Solve the resulting equation to find the value of y: y = -22.
11. Substitute the value of y back into either of the original equations to find the value of x or z. Let's substitute it into the first equation: 10x - 2(-22) - 5z = 0.
12. Solve the resulting equation to find the value of x and z. Plug in the known values to find the value of x: 10x + 44 - 5z = 0. Simplifying gives us 10x - 5z = -44.
13. Substitute the value of x back into any of the original equations to find the value of z. Let's substitute it into the second equation: 8(-4) + 3y + 5z = 10.
14. Solve the resulting equation to find the value of z: -32 + 3y + 5z = 10. Rearranging the equation gives us 3y + 5z = 42.
15. Now we have three equations with three variables: 10x - 5z = -44, 3y + 5z = 42, and 9x + y + 5z = -10.
16. Solve this system of equations using the method of substitution or elimination to find the values of x, y, and z.

Therefore, the solution to the system of equations is x = -4, y = -22, and z = 8.