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The system of equations, x + y + z = 6, x + 2 y + 3 z = 14, x + 3 y + 5z = 20 has
- a)infinitely many solutions
- b)a unique solution
- c)only finitely many solutions
- d)no solution.
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
The system of equations, x + y + z = 6, x + 2 y + 3 z = 14, x + 3 y + ...
The given system of equations does not has a solution if :
0 ⇒ 1(10 -9) - 1(5-3) + 1(3-2)
= 0 ⇒ 1-2 + 1 = 0
Most Upvoted Answer
The system of equations, x + y + z = 6, x + 2 y + 3 z = 14, x + 3 y + ...
Given equations:
The given system of equations is:
1) x + y + z = 6
2) x + 2y + 3z = 14
3) x + 3y + 5z = 20
Method of solving:
To solve this system of equations, we can use the method of elimination.
Step 1: Multiply equations for elimination:
To eliminate x from equations 2 and 3, we can multiply equation 1 by -1 and add it to equations 2 and 3 respectively. This will result in new equations without x.
Multiplying equation 1 by -1, we get:
-1(x + y + z) = -1(6)
- x - y - z = -6
Now, the new system of equations becomes:
4) -x - y - z = -6
5) x + 2y + 3z = 14
6) x + 3y + 5z = 20
Step 2: Eliminate x:
Adding equations 4 and 5, we get:
(-x - y - z) + (x + 2y + 3z) = (-6) + 14
y + 2z = 8
Adding equations 4 and 6, we get:
(-x - y - z) + (x + 3y + 5z) = (-6) + 20
2y + 4z = 14
Step 3: Solve the resulting equations:
Now, we have two equations:
y + 2z = 8 ...(7)
2y + 4z = 14 ...(8)
Step 4: Further simplification:
Dividing equation (8) by 2, we get:
y + 2z = 7
Comparing equations (7) and (8), we can see that they represent the same line. Therefore, they are dependent equations.
Conclusion:
Since the system of equations has dependent equations, it means that the equations are consistent and have infinitely many solutions. Hence, the correct answer is option 'a) infinitely many solutions'.
The given system of equations is:
1) x + y + z = 6
2) x + 2y + 3z = 14
3) x + 3y + 5z = 20
Method of solving:
To solve this system of equations, we can use the method of elimination.
Step 1: Multiply equations for elimination:
To eliminate x from equations 2 and 3, we can multiply equation 1 by -1 and add it to equations 2 and 3 respectively. This will result in new equations without x.
Multiplying equation 1 by -1, we get:
-1(x + y + z) = -1(6)
- x - y - z = -6
Now, the new system of equations becomes:
4) -x - y - z = -6
5) x + 2y + 3z = 14
6) x + 3y + 5z = 20
Step 2: Eliminate x:
Adding equations 4 and 5, we get:
(-x - y - z) + (x + 2y + 3z) = (-6) + 14
y + 2z = 8
Adding equations 4 and 6, we get:
(-x - y - z) + (x + 3y + 5z) = (-6) + 20
2y + 4z = 14
Step 3: Solve the resulting equations:
Now, we have two equations:
y + 2z = 8 ...(7)
2y + 4z = 14 ...(8)
Step 4: Further simplification:
Dividing equation (8) by 2, we get:
y + 2z = 7
Comparing equations (7) and (8), we can see that they represent the same line. Therefore, they are dependent equations.
Conclusion:
Since the system of equations has dependent equations, it means that the equations are consistent and have infinitely many solutions. Hence, the correct answer is option 'a) infinitely many solutions'.
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The system of equations, x + y + z = 6, x + 2 y + 3 z = 14, x + 3 y + 5z = 20 hasa)infinitely many solutionsb)a unique solutionc)only finitely many solutionsd)no solution.Correct answer is option 'D'. Can you explain this answer?
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