{3x+y-4z=-30
{3x+2y+2z=-8
{5x+5y+z=-26
(Give the answer to each equation)
To solve this system of equations, we can use the method of elimination.
First, let's eliminate the variable x by multiplying the first equation by -1 and adding it to the second equation:
-1(3x + y - 4z) = -1(-30)
-3x - y + 4z = 30
3x + 2y + 2z = -8
Adding the two equations together eliminates x:
0 + y + 6z = 22
y + 6z = 22
Next, let's eliminate x again by multiplying the first equation by -5 and adding it to the third equation:
-5(3x + y - 4z) = -5(-30)
-15x - 5y + 20z = 150
5x + 5y + z = -26
Adding the two equations together eliminates x:
0 - 10y + 21z = 124
-10y + 21z = 124
Now we have a system of two equations with two variables:
y + 6z = 22
-10y + 21z = 124
We can solve this system using substitution or elimination. For simplicity, let's use substitution:
Solve the first equation for y:
y = 22 - 6z
Substitute this value of y into the second equation:
-10(22 - 6z) + 21z = 124
-220 + 60z + 21z = 124
81z = 344
z = 344/81
Substitute this value of z into the first equation to solve for y:
y + 6(344/81) = 22
y + 2064/81 = 22
y = 22 - 2064/81
Finally, substitute the values of y and z into any of the original equations to solve for x. Let's use the first equation:
3x + (22 - 2064/81) - 4(344/81) = -30
3x + 22 - 2064/81 - 1376/81 = -30
3x - 3440/81 = -30
3x = -30 + 3440/81
x = (-30 + 3440/81)/3
So the solutions to the system of equations are:
x = (-30 + 3440/81)/3
y = 22 - 2064/81
z = 344/81
To solve this system of equations, we will use the method of elimination.
Step 1: Multiply the first equation by 3 and the second equation by -3 so that the x terms will cancel out when we add the equations together:
Equation 1: 3(3x+y-4z)=-30
9x+3y-12z=-90
Equation 2: -3(3x+2y+2z)=-3(-8)
-9x-6y-6z=24
Step 2: Multiply the third equation by 2, so that when we add it to the sum of the previous two equations, the y terms will cancel out:
Equation 3: 2(5x+5y+z) = 2(-26)
10x+10y+2z = -52
Step 3: Add the three equations together:
(9x+3y-12z)+(-9x-6y-6z)+(10x+10y+2z) = (-90)+(24)+(-52)
This simplifies to:
9x + (-9x) + 10x + 3y + (-6y) + 10y + (-12z) + (-6z) + 2z = -90 + 24 - 52
The like terms combine:
9x - 9x + 10x + 3y - 6y + 10y - 12z - 6z + 2z = -90 + 24 - 52
This further simplifies to:
10x + 7y - 16z = -118
So, the answer to the system is:
10x + 7y - 16z = -118
To find the solution to this system of equations, we can use the method of elimination or substitution. Let's use the method of elimination to solve this system step-by-step.
Step 1: Multiply equation 1 by 5
5(3x + y - 4z) = 5(-30)
15x + 5y - 20z = -150
Step 2: Multiply equation 2 by 3
3(3x + 2y + 2z) = 3(-8)
9x + 6y + 6z = -24
Step 3: Multiply equation 3 by 3
3(5x + 5y + z) = 3(-26)
15x + 15y + 3z = -78
The system becomes:
15x + 5y - 20z = -150
9x + 6y + 6z = -24
15x + 15y + 3z = -78
Step 4: Multiply equation 2 by -5
-5(9x + 6y + 6z) = -5(-24)
-45x - 30y - 30z = 120
Step 5: Add equation 4 to equation 1
(15x + 5y - 20z) + (-45x - 30y - 30z) = (-150) + 120
-30x - 25z = -30
Step 6: Multiply equation 3 by -3
-3(15x + 15y + 3z) = -3(-78)
-45x - 45y - 9z = 234
Step 7: Add equation 6 to equation 1
(-30x - 25z) + (-45x - 45y - 9z) = (-30) + 234
-75x - 45y - 34z = 204
The system becomes:
-75x - 45y - 34z = 204
9x + 6y + 6z = -24
15x + 15y + 3z = -78
Step 8: Multiply equation 2 by 5
5(9x + 6y + 6z) = 5(-24)
45x + 30y + 30z = -120
Step 9: Add equation 8 to equation 1
(-75x - 45y - 34z) + (45x + 30y + 30z) = 204 + (-120)
-30x - 4z = 84
Step 10: Multiply equation 3 by -2
-2(15x + 15y + 3z) = -2(-78)
-30x - 30y - 6z = 156
Step 11: Add equation 10 to equation 2
(9x + 6y + 6z) + (-30x - 30y - 6z) = -24 + 156
-21x = 132
x = -132/21
x = -6
Step 12: Substitute x = -6 into equation 2
9(-6) + 6y + 6z = -24
-54 + 6y + 6z = -24
6y + 6z = -24 + 54
6y + 6z = 30
y + z = 5 (Dividing the equation by 6)
Step 13: Substitute the value of x from step 11 into equation 3
15(-6) + 15y + 3z = -78
-90 + 15y + 3z = -78
15y + 3z = 12
5y + z = 4 (Dividing the equation by 3)
Since we have simplified equations in terms of y + z and 5y + z, we can equate these two expressions:
y + z = 5
5y + z = 4
Step 14: Subtract the first equation from the second equation
(5y + z) - (y + z) = 4 - 5
4y = -1
y = -1/4
Step 15: Substitute the value of y into the equation y + z = 5
-1/4 + z = 5
z = 5 + 1/4
z = 5.25
Therefore, the solution to the system of equations is:
x = -6, y = -1/4, z = 5.25