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The System of equations,
x + y + z = 8
x - y + 2z = 6
3x + 5y+ 7z= 14 has,
- a)no solution
- b)Unique Solution
- c)infinite no. of Solutions
- d)finite no. of solutions
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
The System of equations,x + y + z = 8x - y + 2z = 63x + 5y+ 7z= 14 has...
The augmented matrix C = [A : B]
Rank of A = 3, rank of C = 3.
So, rank of A = rank of C = 3 = number of unknowns. Hence the equations are consistent with unique solution.
So, rank of A = rank of C = 3 = number of unknowns. Hence the equations are consistent with unique solution.
Most Upvoted Answer
The System of equations,x + y + z = 8x - y + 2z = 63x + 5y+ 7z= 14 has...
To solve the system of equations:
x + y + z = 8x - y
2z = 6
x + 5y + 7z = 14
We can start by solving the second equation, which is a simple equation with one variable. From the equation 2z = 6, we can solve for z by dividing both sides by 2: z = 3.
Now, let's substitute this value of z into the first and third equations to simplify the system further.
Substituting z = 3 into the first equation:
x + y + 3 = 8x - y
Rearranging the equation, we get:
9x - 2y = 3
Now, substituting z = 3 into the third equation:
x + 5y + 7(3) = 14
x + 5y + 21 = 14
x + 5y = -7
So, we have two new equations:
9x - 2y = 3
x + 5y = -7
Now, we can solve this system of equations using any method, such as substitution or elimination. Let's use the elimination method to solve for x and y.
Multiplying the second equation by 9, we get:
9x + 45y = -63
Now, we can add this equation to the first equation to eliminate x:
(9x - 2y) + (9x + 45y) = 3 + (-63)
18x + 43y = -60
Now, we have two equations:
18x + 43y = -60
x + 5y = -7
This is a system of linear equations with two variables. By solving this system, we can find the values of x and y.
Solving this system of equations, we find a unique solution for x and y:
x = -3
y = 2
Finally, we can substitute these values back into any of the original equations to find the value of z.
Using the second equation:
2z = 6
z = 3
So, the solution to the system of equations is:
x = -3
y = 2
z = 3
Therefore, the correct answer is option B - Unique Solution.
x + y + z = 8x - y
2z = 6
x + 5y + 7z = 14
We can start by solving the second equation, which is a simple equation with one variable. From the equation 2z = 6, we can solve for z by dividing both sides by 2: z = 3.
Now, let's substitute this value of z into the first and third equations to simplify the system further.
Substituting z = 3 into the first equation:
x + y + 3 = 8x - y
Rearranging the equation, we get:
9x - 2y = 3
Now, substituting z = 3 into the third equation:
x + 5y + 7(3) = 14
x + 5y + 21 = 14
x + 5y = -7
So, we have two new equations:
9x - 2y = 3
x + 5y = -7
Now, we can solve this system of equations using any method, such as substitution or elimination. Let's use the elimination method to solve for x and y.
Multiplying the second equation by 9, we get:
9x + 45y = -63
Now, we can add this equation to the first equation to eliminate x:
(9x - 2y) + (9x + 45y) = 3 + (-63)
18x + 43y = -60
Now, we have two equations:
18x + 43y = -60
x + 5y = -7
This is a system of linear equations with two variables. By solving this system, we can find the values of x and y.
Solving this system of equations, we find a unique solution for x and y:
x = -3
y = 2
Finally, we can substitute these values back into any of the original equations to find the value of z.
Using the second equation:
2z = 6
z = 3
So, the solution to the system of equations is:
x = -3
y = 2
z = 3
Therefore, the correct answer is option B - Unique Solution.
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