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The equations 2x + 3y + 4 = 0; 3x + 4y + 6 = 0 and 4x + 5y + 8 = 0 are
- a)consistent with unique solution
- b)in consistent
- c)consistent with infinitely many solutions
- d)none of the above
Correct answer is option 'A'. Can you explain this answer?
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Most Upvoted Answer
The equations 2x + 3y + 4 = 0; 3x + 4y + 6 = 0 and 4x + 5y + 8 = 0 are...
Given equations are:
2x + 3y + 4 = 0
3x + 4y + 6 = 0
4x + 5y + 8 = 0
We need to determine whether the given system of equations is consistent with a unique solution, inconsistent or consistent with infinitely many solutions.
Solution:
To check the consistency of the given system of equations, we need to solve the equations simultaneously.
Step 1: We can eliminate x by multiplying equation (1) by 3 and equation (2) by 2 and subtracting them.
6x + 9y + 12 = 0
6x + 8y + 12 = 0
y = -1
Step 2: Substitute y = -1 in equation (1) to get the value of x.
2x + 3(-1) + 4 = 0
2x = -7
x = -7/2
Therefore, the system of equations has a unique solution.
Hence, the correct option is (a) consistent with unique solution.
Note: If we try to eliminate x by multiplying equation (1) by 4 and equation (3) by 3 and subtracting them, we get a contradiction. Therefore, the system of equations is not inconsistent. Also, if we try to eliminate y by multiplying equation (1) by 5 and equation (2) by 4 and subtracting them or by multiplying equation (2) by 5 and equation (3) by 4 and subtracting them, we get a contradiction. Therefore, the system of equations is not consistent with infinitely many solutions.
2x + 3y + 4 = 0
3x + 4y + 6 = 0
4x + 5y + 8 = 0
We need to determine whether the given system of equations is consistent with a unique solution, inconsistent or consistent with infinitely many solutions.
Solution:
To check the consistency of the given system of equations, we need to solve the equations simultaneously.
Step 1: We can eliminate x by multiplying equation (1) by 3 and equation (2) by 2 and subtracting them.
6x + 9y + 12 = 0
6x + 8y + 12 = 0
y = -1
Step 2: Substitute y = -1 in equation (1) to get the value of x.
2x + 3(-1) + 4 = 0
2x = -7
x = -7/2
Therefore, the system of equations has a unique solution.
Hence, the correct option is (a) consistent with unique solution.
Note: If we try to eliminate x by multiplying equation (1) by 4 and equation (3) by 3 and subtracting them, we get a contradiction. Therefore, the system of equations is not inconsistent. Also, if we try to eliminate y by multiplying equation (1) by 5 and equation (2) by 4 and subtracting them or by multiplying equation (2) by 5 and equation (3) by 4 and subtracting them, we get a contradiction. Therefore, the system of equations is not consistent with infinitely many solutions.
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Community Answer
The equations 2x + 3y + 4 = 0; 3x + 4y + 6 = 0 and 4x + 5y + 8 = 0 are...
Consider first two equations :
2x + 3y = –4 and 3x + 4y = –6
∴ x = –2 and y = 0
Now this solution satisfies the third, so the equations are consistent with unique solution.
2x + 3y = –4 and 3x + 4y = –6
∴ x = –2 and y = 0
Now this solution satisfies the third, so the equations are consistent with unique solution.
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The equations 2x + 3y + 4 = 0; 3x + 4y + 6 = 0 and 4x + 5y + 8 = 0 area)consistent with unique solutionb)in consistentc)consistent with infinitely many solutionsd)none of the aboveCorrect answer is option 'A'. Can you explain this answer?
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