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If x + 2y+ 2z= 1, 2x+y + z = -2, 3x + 2y + 2z = 3, y + z = 0 is a system of equations, then
- a)no solution exists
- b)a unique solution exists
- c)an infinite number of solutions exist
- d)None of the above
Correct answer is option 'B'. Can you explain this answer?
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If x+ 2y+ 2z= 1, 2x+y + z = -2, 3x + 2y + 2z = 3, y + z = 0 is a syste...
To determine the existence and uniqueness of solutions for the given system of equations, we can use the method of elimination or substitution. Let's solve the system step by step:
Step 1: Multiply the second equation by 2 and add it to the first equation to eliminate the y term:
(2x - y + z) + 2(2x + y - z) = -2 + 2(1)
4x = 0
x = 0
Step 2: Substitute the value of x = 0 into the third equation:
3(0) - 2y + 2z = 3
-2y + 2z = 3
Step 3: Multiply the second equation by 2 and add it to the third equation to eliminate the y term:
2(-2y + 2z) + (-2y + 2z) = 2(3) + 3
-2y + 2z - 2y + 2z = 6 + 3
4z = 9
z = 9/4
Step 4: Substitute the values of x = 0 and z = 9/4 into the second equation:
2(0) - y + (9/4) = -2
-y + (9/4) = -2
-y = -2 - (9/4)
-y = -8/4 - 9/4
-y = -17/4
y = 17/4
Therefore, the solution to the system of equations is x = 0, y = 17/4, and z = 9/4.
Explanation:
The given system of equations has a unique solution because after solving the equations step by step, we obtained specific values for each variable (x, y, and z) that satisfy all the equations simultaneously. In other words, there is a single solution that satisfies all the given equations.
Hence, the correct answer is option B) a unique solution exists.
Step 1: Multiply the second equation by 2 and add it to the first equation to eliminate the y term:
(2x - y + z) + 2(2x + y - z) = -2 + 2(1)
4x = 0
x = 0
Step 2: Substitute the value of x = 0 into the third equation:
3(0) - 2y + 2z = 3
-2y + 2z = 3
Step 3: Multiply the second equation by 2 and add it to the third equation to eliminate the y term:
2(-2y + 2z) + (-2y + 2z) = 2(3) + 3
-2y + 2z - 2y + 2z = 6 + 3
4z = 9
z = 9/4
Step 4: Substitute the values of x = 0 and z = 9/4 into the second equation:
2(0) - y + (9/4) = -2
-y + (9/4) = -2
-y = -2 - (9/4)
-y = -8/4 - 9/4
-y = -17/4
y = 17/4
Therefore, the solution to the system of equations is x = 0, y = 17/4, and z = 9/4.
Explanation:
The given system of equations has a unique solution because after solving the equations step by step, we obtained specific values for each variable (x, y, and z) that satisfy all the equations simultaneously. In other words, there is a single solution that satisfies all the given equations.
Hence, the correct answer is option B) a unique solution exists.
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