ii) x + 2y -3z=-1
3x -y + 2z=7
5x + 3y- 4z=2
To solve the system of equations:
ii) x + 2y - 3z = -1
3x - y + 2z = 7
5x + 3y - 4z = 2
We can use any of the common methods for solving systems of equations, such as substitution, elimination, or matrix operations. Let's use the method of elimination:
Step 1: Multiply the equations to create coefficients that will cancel each other out. We want to create opposite coefficients for either the x, y, or z term in two of the equations.
Multiply equation ii) by 3, equation iii) by 2, and equation i) by 5, resulting in:
iii) 3(x + 2y - 3z) = 3(-1x + 2y - 3z) = -3x + 6y - 9z = -3
i) 5(3x - y + 2z) = 5(7x - y + 2z) = 15x - 5y + 10z = 35
ii) 2(5x + 3y - 4z) = 2(2x + 3y - 4z) = 10x + 6y - 8z = 4
Step 2: Add the resulting equations together. This will eliminate one variable, leaving you with two equations in two variables.
-3x + 6y - 9z + (15x - 5y + 10z) + (10x + 6y - 8z) = -3 + 35 + 4
Simplifying:
-3x + 15x + 10x + 6y - 5y + 6y - 9z + 10z - 8z = 36
Combine like terms:
22x + 7y - 11z = 36
Step 3: Repeat steps 1 and 2 with different pairs of equations to eliminate a different variable. Let's use equations ii) and iii) this time.
Multiply equation iii) by 3 and equation ii) by 5, resulting in:
iii) 3(-3x + 6y - 9z) = -9x + 18y - 27z = -9
ii) 5(2x + 3y - 4z) = 10x + 15y - 20z = 10
Adding these new equations together:
-9x + 18y - 27z + (10x + 15y - 20z) = -9 + 10
Simplifying:
-9x + 10x + 18y + 15y - 27z - 20z = 1
Combine like terms:
x + 33y - 47z = 1
Now we have a new equation in terms of x, y, and z:
22x + 7y - 11z = 36
x + 33y - 47z = 1
Step 4: Solve for one variable in terms of the other variables in either equation.
Let's solve for x in terms of y and z in the second equation:
x = 1 - 33y + 47z
Step 5: Substitute this expression for x into the first equation:
22(1 - 33y + 47z) + 7y - 11z = 36
Simplify and rearrange terms:
22 - 726y + 1034z + 7y - 11z = 36
Combine like terms:
-719y + 1023z = 14
Now we have a new equation with only y and z:
-719y + 1023z = 14
Step 6: Solve for one variable in terms of the other variable in this equation. Let's solve for y in terms of z:
y = (14 - 1023z)/(-719)
Step 7: Substitute this expression for y into the equation from Step 6:
x = 1 - 33[(14 - 1023z)/(-719)] + 47z
Simplify and rearrange terms as needed to find the solution for x.
x = (75239z + 100331)/719
Now we have found the values of x, y, and z in terms of z. We can substitute these expressions into any of the original equations to find the exact values for x, y, and z.