How do i solve inequalitites with three variables. I still can't figure it outplease help.
x+y=1
y-z=-3
2x+3y+z=1
the answers are 3,-2,1 how do get there by solving linear equations with three variables.
5x-2y-z=2
-y+2z=11
x+y+z=-3
To solve inequalities with three variables, you can use a method known as substitution or elimination. Let's go through the steps for each system of equations you provided.
System of Equations 1:
1. Begin by isolating one variable in one of the equations. Let's isolate x in the first equation: x = 1 - y.
2. Substitute this value of x into the other two equations to eliminate x.
- Substituting x in the second equation: (1 - y) - z = -3.
- Substituting x in the third equation: 2(1 - y) + 3y + z = 1.
3. Simplify and solve the resulting linear equations:
- Rearrange the second equation to have a z term: 1 - y - z = -3.
- Combine similar terms to get: -y - z = -4. (Equation 4)
- Rearrange the third equation to have a y term: 2 - 2y + 3y + z = 1.
- Combine similar terms to get: y + z = -1. (Equation 5)
4. Now, you have equations 4 and 5 with two variables. Solve them simultaneously.
- Multiply Equation 5 by -1: -y - z = -1. (Equation 6)
- Add Equation 4 to Equation 6 to eliminate y: 0 = -5.
Since there is no valid solution to 0 = -5, this system of equations is inconsistent. Therefore, there is no solution.
System of Equations 2:
1. Begin by isolating one variable in one of the equations. Let's isolate x in the third equation: x = -y - z - 3.
2. Substitute this value of x into the other two equations to eliminate x.
- Substituting x in the first equation: 5(-y - z - 3) - 2y - z = 2.
- Substituting x in the second equation: -(-y - z - 3) + 2z = 11.
3. Simplify and solve the resulting linear equations:
- Distribute and combine similar terms in the first equation: -7y - 6z = -13. (Equation 7)
- Simplify and combine similar terms in the second equation: y + 3z = -8. (Equation 8)
4. Now, you have equations 7 and 8 with two variables. Solve them simultaneously.
- Multiply Equation 8 by 7: 7y + 21z = -56. (Equation 9)
- Add Equation 9 to Equation 7 to eliminate y: 15z = -69.
- Simplify Equation 9 to: 7y = -56 - 21z.
- Substitute the value of z back into Equation 7 to solve for y: -7y = 39.
- Solve for y: y = 39/(-7) = -(-39/7) = 39/7.
- Substitute the values of y and z back into the equation x = -y - z - 3 to solve for x: x = -39/7 - 39/7 - 63/7 = -3 - 6 - 9 = -18/7.
So the solution for this system of equations is x = -18/7, y = 39/7, z = -69/15 = -23/5.
I hope this helps! Let me know if you have any other questions.