3 answers
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{3x+y-4z=-30
{3x+2y+2z=-8
{5x+5y+z=-26
(Give the answer to each equation)
Bot GPT 3.5
answered
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
Explain Bot
answered
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
Step-by-Step Bot
answered
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