What is the solution of the system of equations?
-3x-4y-3z=-7
2x-6y+2z=3
5x-2y+5z=9
A. (5, –2, 7)
B. (–5, 2, 7)
C. (5, 2, –7)
D. no solution
To solve the system of equations, we can use the method of elimination or substitution. Let's use the method of elimination.
First, we need to eliminate either the x, y, or z term. Let's eliminate the x term by multiplying the first equation by 2 and adding it to the second equation:
2*(-3x-4y-3z) + (2x-6y+2z) = 2*(-7) + 3
-6x - 8y - 6z + 2x - 6y + 2z = -14 + 3
-4x - 14y - 4z = -11
Now, let's eliminate the x term in the third equation as well:
2*(5x-2y+5z) + (-3*(-3x-4y-3z)) = 2*9 + 7
10x - 4y + 10z + 9x + 12y + 9z = 18 + 7
19x + 8y + 19z = 25
Now we have a system of two equations with two variables:
-4x - 14y - 4z = -11
19x + 8y + 19z = 25
We can solve this system of equations by elimination as well.
First, let's eliminate the y term by multiplying the first equation by 8 and the second equation by 14:
8*(-4x - 14y - 4z) + 14*(19x + 8y + 19z) = 8*(-11) + 14*25
-32x - 112y - 32z + 266x + 112y + 266z = -88 + 350
234x + 234z = 262
Now, let's solve for x and z by dividing the equation by 234:
x + z = 262/234
x + z = 1.12
Now, let's go back to one of the original equations to solve for y. Let's use the first equation:
-3x - 4y - 3z = -7
Since we know that x + z = 1.12, we can substitute this into the equation:
-3(1.12) - 4y - 3(1.12) = -7
-3.36 - 4y - 3.36 = -7
-4y - 6.72 = -7
-4y = -7 + 6.72
-4y = -0.28
y = 0.07
So the solution to the system of equations is (x, y, z) = (1.12, 0.07, 1.12).
None of the given options match this solution, so the answer is D. no solution.
To find the solution of the system of equations, we will use the method of elimination.
First, let's multiply the first equation by 2 and the second equation by 3 so that the coefficients of x becomes opposite in sign. This will allow us to eliminate x when we add the two equations together.
Equation 1 (multiplied by 2): -6x - 8y - 6z = -14
Equation 2 (multiplied by 3): 6x - 18y + 6z = 9
Now add the two equations together:
(-6x + 6x) + (-8y - 18y) + (-6z + 6z) = -14 + 9
-26y = -5
Divide by -26 to solve for y:
y = (-5) / (-26)
y = 5/26
Now substitute the value of y in any of the original equations. Let's use the first equation:
-3x - 4(5/26) - 3z = -7
Multiply through by 26 to eliminate the fraction:
-78x - 20 - 78z = -182
Rearrange the terms:
-78x - 78z = -182 + 20
-78x - 78z = -162
Divide through by -78 to solve for x:
x = (-162) / (-78)
x = 9/13
Now substitute the values of x and y in any of the original equations. Let's use the second equation:
2(9/13) - 6(5/26) + 2z = 3
Multiply through by 13 to eliminate the fraction:
18 - 15 + 26z = 39
Combine like terms:
26z = 36
Divide by 26 to solve for z:
z = 36 / 26
z = 18 / 13
Therefore, the solution to the system of equations is (x, y, z) = (9/13, 5/26, 18/13).
The closest answer choice is C. (5, 2, -7), but it is not an exact match, so the correct answer is D. no solution.
To find the solution of the system of equations, we need to solve the system by using one of the methods such as elimination, substitution, or Gaussian elimination.
Let's use the elimination method to solve the system of equations:
-3x - 4y - 3z = -7 --> Equation 1
2x - 6y + 2z = 3 --> Equation 2
5x - 2y + 5z = 9 --> Equation 3
First, let's eliminate the variable z. Multiply Equation 2 by 3 and Equation 1 by 2, then add them:
6x - 18y + 6z = 9 --> Multiply Equation 2 by 3
-6x - 8y - 6z = -14 --> Multiply Equation 1 by 2
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-26y = -5 --> Add the equations
Dividing both sides of the resulting equation by -26, we get:
y = -5 / -26
y = 5/26
Now, substitute the value of y = 5/26 into Equation 1 or Equation 2 to solve for x:
-3x - 4(5/26) - 3z = -7 --> Equation 1, substitute y = 5/26
-3x - 20/26 - 3z = -7
-3x - 20/26 - 78z/26 = -7
-78x - 20 - 78z = -182 --> Multiply through by 26 to eliminate fraction
Similarly, substitute the value of y = 5/26 into Equation 3 to solve for z:
5x - 2(5/26) + 5z = 9 --> Equation 3, substitute y = 5/26
5x - 10/26 + 5z = 9
130x - 10 - 130z = 234 --> Multiply through by 26 to eliminate fraction
Now, we have a system of two equations with two variables:
-78x - 78z = -162 -->(Equation 4)
130x - 130z = 244 -->(Equation 5)
We can solve this system using the elimination method. Multiply Equation 4 by -5 and Equation 5 by 4, then add them:
390x + 390z = 810 --> Multiply Equation 4 by -5
520x - 520z = 976 --> Multiply Equation 5 by 4
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910x = 1786 --> Add the equations
Dividing both sides of the resulting equation by 910, we get:
x = 1786 / 910
x = 1786 / (2 * 455)
x = 893 / 455
x = 19 / 13
Now, substitute the value of x = 19/13 into Equation 4 to solve for z:
-78(19/13) - 78z = -162 --> Equation 4, substitute x = 19/13
-(78 * 19) / 13 - 78z = -162
-1482 / 13 - 1014z / 13 = -162
-1482 - 1014z = -2106 --> Multiply through by 13 to eliminate fraction
Now, let's solve for z:
-1014z = -624 --> Subtract -1482 from both sides
z = -624 / -1014
z = 312 / 507
z = 104 / 169
Therefore, the solution of the system of equations is (x, y, z) = (19/13, 5/26, 104/169).
Comparing this solution to the given answer choices, none of the options matches the solution. Therefore, the correct answer would be D. no solution.