trying to solve linear equations problem is: x-y-2z=4; -x+2y+z=1, -x+y-3z=11. please help me.
To solve the given system of linear equations, you can use the method of elimination or substitution. I will demonstrate both methods so you can choose the one you prefer.
Method 1: Elimination Method
Step 1: Multiply the equations by appropriate constants to make the coefficients of x in the first two equations cancel each other when added, and do the same for y in the first and third equations.
- Equation 1: x - y - 2z = 4 (left as it is)
- Equation 2: -x + 2y + z = 1 (multiply by 2 to cancel x with Equation 1)
- Equation 3: -x + y - 3z = 11 (multiply by -1 to cancel x with Equation 1)
The modified equations become:
- Equation 1: x - y - 2z = 4
- Equation 2: -2x + 4y + 2z = 2
- Equation 3: x - y + 3z = -11
Step 2: Add the modified equations to eliminate x and solve for y.
Adding Equation 1 and Equation 3:
(x - y - 2z) + (x - y + 3z) = 4 + (-11)
2x - 2y + z = -7
Step 3: Add the resulting equation to Equation 2 to eliminate y and solve for z.
Adding Equation 2 and the equation obtained in Step 2:
(-2x + 4y + 2z) + (2x - 2y + z) = 2 + (-7)
3z = -5
Therefore, z = -5/3.
Step 4: Substitute the value of z into any of the original equations to solve for y.
Using Equation 1:
x - y - 2(-5/3) = 4
x - y + 10/3 = 4
x - y = 4 - 10/3
x - y = 2/3
Therefore, x - y = 2/3.
Step 5: Substitute the values of z and x obtained into any of the original equations to solve for y.
Using Equation 2:
-x + 2y + (-5/3) = 1
-1 + 2y - 5/3 = 1
2y - 5/3 = 2
Simplifying the equation:
2y = 2 + 5/3
2y = 11/3
y = (11/3) / 2
y = 11/6
Hence, the solution to the system of equations is x = 2/3, y = 11/6, and z = -5/3.
Method 2: Substitution Method
Step 1: Choose one equation and solve it for one variable in terms of the other variables.
From Equation 2, we can solve for x:
-x = 1 - 2y - z
x = -1 + 2y + z
Step 2: Substitute the value of x obtained above into the other equations.
Substituting x into Equation 1 and Equation 3:
-1 + 2y + z - y - 2z = 4 (Equation 1)
-1 + 2y + z - y + 3z = 11 (Equation 3)
Simplifying the equations:
y - z = 5 (Equation A)
y + 4z = 12 (Equation B)
Step 3: Solve the resulting system of equations (Equation A and Equation B).
Multiply Equation A by 4 and subtract it from Equation B to eliminate y:
4y - 4z = 20
y + 4z = 12
____________________
-3y = -8
y = 8/3
Substituting the value of y back into Equation A:
8/3 - z = 5
z = -(13/3)
Step 4: Substitute the values of y and z obtained into the expression for x.
From the expression for x:
x = -1 + 2y + z
x = -1 + 2(8/3) - (13/3)
x = -1 + 16/3 - 13/3
x = -(1/3)
Therefore, the solution to the system of equations is x = -(1/3), y = 8/3, and z = -(13/3).