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The system of equations x − 4y + 7z = 14, 3x + 8y − 2z = 13, 7x − 8y + 26z = 5 has
- a)a unique solution
- b)no solution
- c)an infinite number of solution
- d)none of these
Correct answer is option 'B'. Can you explain this answer?
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Most Upvoted Answer
The system of equations x − 4y + 7z = 14, 3x + 8y − 2z = 13, 7x − 8y ...
Thus system is inconsistent i.e. has no solution.
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The system of equations x − 4y + 7z = 14, 3x + 8y − 2z = 13, 7x − 8y ...
Given system of equations:
1) x - 4y + 7z = 14
2) 3x - 8y - 2z = 13
3) 7x - 8y + 26z = 5
To determine if the system of equations has a unique solution, no solution, or an infinite number of solutions, we can use the method of elimination or substitution.
Method of Elimination:
1) Multiply equation 1 by 3 and equation 2 by 1 to eliminate x:
3(x - 4y + 7z) = 3(14) -> 3x - 12y + 21z = 42
1(3x - 8y - 2z) = 1(13) -> 3x - 8y - 2z = 13
2) Subtract equation 2 from equation 1 to eliminate x:
(3x - 12y + 21z) - (3x - 8y - 2z) = 42 - 13
-4y + 23z = 29
3) Multiply equation 1 by 7 and equation 3 by 1 to eliminate x:
7(x - 4y + 7z) = 7(14) -> 7x - 28y + 49z = 98
1(7x - 8y + 26z) = 1(5) -> 7x - 8y + 26z = 5
4) Subtract equation 3 from equation 1 to eliminate x:
(7x - 28y + 49z) - (7x - 8y + 26z) = 98 - 5
-20y + 23z = 93
Now we have the following two equations:
-4y + 23z = 29
-20y + 23z = 93
Method of Substitution:
From equation 1, we can express x in terms of y and z:
x = 4y - 7z + 14
Substituting this expression for x in equations 2 and 3, we get:
3(4y - 7z + 14) - 8y - 2z = 13 -> 12y - 21z + 42 - 8y - 2z = 13
7(4y - 7z + 14) - 8y + 26z = 5 -> 28y - 49z + 98 - 8y + 26z = 5
Simplifying these equations, we get:
4y - 23z = -25
20y - 23z = -93
Conclusion:
We now have the following two equations:
-4y + 23z = 29
-4y + 23z = -25
By comparing the coefficients of y and z, we can see that the equations are inconsistent. The left side of the equations are the same, but the right sides are different. Therefore, the system of equations has no solution (option B).
1) x - 4y + 7z = 14
2) 3x - 8y - 2z = 13
3) 7x - 8y + 26z = 5
To determine if the system of equations has a unique solution, no solution, or an infinite number of solutions, we can use the method of elimination or substitution.
Method of Elimination:
1) Multiply equation 1 by 3 and equation 2 by 1 to eliminate x:
3(x - 4y + 7z) = 3(14) -> 3x - 12y + 21z = 42
1(3x - 8y - 2z) = 1(13) -> 3x - 8y - 2z = 13
2) Subtract equation 2 from equation 1 to eliminate x:
(3x - 12y + 21z) - (3x - 8y - 2z) = 42 - 13
-4y + 23z = 29
3) Multiply equation 1 by 7 and equation 3 by 1 to eliminate x:
7(x - 4y + 7z) = 7(14) -> 7x - 28y + 49z = 98
1(7x - 8y + 26z) = 1(5) -> 7x - 8y + 26z = 5
4) Subtract equation 3 from equation 1 to eliminate x:
(7x - 28y + 49z) - (7x - 8y + 26z) = 98 - 5
-20y + 23z = 93
Now we have the following two equations:
-4y + 23z = 29
-20y + 23z = 93
Method of Substitution:
From equation 1, we can express x in terms of y and z:
x = 4y - 7z + 14
Substituting this expression for x in equations 2 and 3, we get:
3(4y - 7z + 14) - 8y - 2z = 13 -> 12y - 21z + 42 - 8y - 2z = 13
7(4y - 7z + 14) - 8y + 26z = 5 -> 28y - 49z + 98 - 8y + 26z = 5
Simplifying these equations, we get:
4y - 23z = -25
20y - 23z = -93
Conclusion:
We now have the following two equations:
-4y + 23z = 29
-4y + 23z = -25
By comparing the coefficients of y and z, we can see that the equations are inconsistent. The left side of the equations are the same, but the right sides are different. Therefore, the system of equations has no solution (option B).
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The system of equations x − 4y + 7z = 14, 3x + 8y − 2z = 13, 7x − 8y + 26z = 5 hasa)a unique solutionb)no solutionc)an infinite number of solutiond)none of theseCorrect answer is option 'B'. Can you explain this answer?
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