Solve for
3x + 6Y - 9z = 3
5x - 7y + 10z = -4
11x +4y - 6z = 2
take the first equation, and the third equation.
mulitiply the first by 2, and the third by 3.
6x+12y-18z=6
33x+12y-18z=6
that implies that x is zero.
then the first and second
6y-9z=3
-7y+10z=-4
multply the first by 10, the second by 9
60y-90z=30
-63y+90z=-36
add,
-3y=6 or y=-2
solve for z in any equation, check your answers.
Since you don't specify which method, let's use good ol' elimination.
#1x2 ---> 6x+12y-18z=6
#3x3 ---> 33x+12y-18z=6
subtract, by luck both z and y disappear
x = 0
#2x3 ---> 15x-21y+30z=-12
#3x5 ---> 55x+20y-30z=10
add
70x-y=-2 but x=0, so
y = 2
sub back into #3 to get z = 1
x=0
y=2
z=1
2x^3(2x^2+4x+3)
To solve the system of equations:
3x + 6y - 9z = 3 ......(Equation 1)
5x - 7y + 10z = -4 ......(Equation 2)
11x + 4y - 6z = 2 ......(Equation 3)
There are multiple methods to solve this system, but one common approach is to use the method of substitution. Here's how you can do it:
Step 1: Solve Equation 1 for x
3x = -6y + 9z + 3
x = (-6y + 9z + 3) / 3
x = -2y + 3z + 1 ......(Equation 4)
Step 2: Substitute Equation 4 into Equations 2 and 3
5(-2y + 3z + 1) - 7y + 10z = -4 ......(Equation 5)
11(-2y + 3z + 1) + 4y - 6z = 2 ......(Equation 6)
Simplify Equations 5 and 6:
-10y + 15z + 5 - 7y + 10z = -4 (expand the multiplication)
-17y + 25z + 5 = -4
-17y + 25z = -9 ......(Equation 7)
-22y + 33z + 11 + 4y - 6z = 2 (expand the multiplication)
-18y + 27z + 11 = 2
-18y + 27z = -9 ......(Equation 8)
Step 3: Solve Equations 7 and 8 as a system of linear equations
Let's solve the system of equations using the method of elimination. Multiply Equation 7 by 3 and Equation 8 by 25 to eliminate the y term:
-51y + 75z = -27 ......(Equation 9)
-450y + 675z = -225 ......(Equation 10)
Now we can subtract Equation 9 from Equation 10 to eliminate the y term:
(-450y + 675z) - (-51y + 75z) = -225 - (-27)
-450y + 675z + 51y - 75z = -225 + 27
-399y + 600z = -198 ......(Equation 11)
Step 4: Solve Equation 11 for z
z = (-198 + 399y) / 600
Step 5: Substitute the value of z into Equation 7 or 8 to solve for y
-17y + 25((-198 + 399y) / 600) = -9
Now, solve this equation for y.
After finding the value of y, substitute it back into Equation 4 to solve for x.
Finally, substitute the values of x, y, and z back into any of the original equations (Equations 1, 2, or 3) to check if the solution is consistent.
This step-by-step method of substitution and elimination helps in finding the solution for a system of linear equations. Note that the calculations involved might be complex, but it's essential to go through each step accurately to arrive at the correct solution.