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If x=a, y=b, z=c is a solution of the system of linear equations x +8y +7z=0 9x +2y +3z=0 x +y +z=0 such that the point(a,b,c) lies on plane x +2y +z=6 then 2a +b +c equals to a.-1 b.0 c.1 d.2?
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If x=a, y=b, z=c is a solution of the system of linear equations x +8y...
Understanding the System of Equations
To find the solution of the system of equations:
1. Equations:
- \( x + 8y + 7z = 0 \)
- \( 9x + 2y + 3z = 0 \)
- \( x + y + z = 0 \)
We can express \( z \) in terms of \( x \) and \( y \) from the third equation:
- \( z = -x - y \)
Substituting this into the first two equations allows us to solve for \( x \) and \( y \).
Substituting z into the First Equation
- \( x + 8y + 7(-x - y) = 0 \)
- Simplifying, we get:
- \( x + 8y - 7x - 7y = 0 \)
- \( -6x + y = 0 \)
- Thus, \( y = 6x \)
Substituting into the Second Equation
- Replacing \( y \) in the second equation:
- \( 9x + 2(6x) + 3(-x - 6x) = 0 \)
- Simplifying leads to:
- \( 9x + 12x - 21x = 0 \)
- Thus, \( 0 = 0 \), confirming dependent equations.
Finding Coordinates (a, b, c)
Using \( y = 6x \) and \( z = -7x \):
- \( a = x \)
- \( b = 6x \)
- \( c = -7x \)
Condition for the Plane
The point \( (a, b, c) \) lies on the plane \( x + 2y + z = 6 \):
- Substituting gives:
- \( x + 2(6x) - 7x = 6 \)
- Simplifying leads to:
- \( 6 = 6 \), confirming all points satisfy the plane.
Calculating \( 2a + b + c \)
- \( 2a + b + c = 2x + 6x - 7x = x \)
Hence, we know that:
- \( x = 1 \) satisfies the equations.
Final Value
Thus, \( 2a + b + c = 1 \). The answer is c. 1.
To find the solution of the system of equations:
1. Equations:
- \( x + 8y + 7z = 0 \)
- \( 9x + 2y + 3z = 0 \)
- \( x + y + z = 0 \)
We can express \( z \) in terms of \( x \) and \( y \) from the third equation:
- \( z = -x - y \)
Substituting this into the first two equations allows us to solve for \( x \) and \( y \).
Substituting z into the First Equation
- \( x + 8y + 7(-x - y) = 0 \)
- Simplifying, we get:
- \( x + 8y - 7x - 7y = 0 \)
- \( -6x + y = 0 \)
- Thus, \( y = 6x \)
Substituting into the Second Equation
- Replacing \( y \) in the second equation:
- \( 9x + 2(6x) + 3(-x - 6x) = 0 \)
- Simplifying leads to:
- \( 9x + 12x - 21x = 0 \)
- Thus, \( 0 = 0 \), confirming dependent equations.
Finding Coordinates (a, b, c)
Using \( y = 6x \) and \( z = -7x \):
- \( a = x \)
- \( b = 6x \)
- \( c = -7x \)
Condition for the Plane
The point \( (a, b, c) \) lies on the plane \( x + 2y + z = 6 \):
- Substituting gives:
- \( x + 2(6x) - 7x = 6 \)
- Simplifying leads to:
- \( 6 = 6 \), confirming all points satisfy the plane.
Calculating \( 2a + b + c \)
- \( 2a + b + c = 2x + 6x - 7x = x \)
Hence, we know that:
- \( x = 1 \) satisfies the equations.
Final Value
Thus, \( 2a + b + c = 1 \). The answer is c. 1.
Community Answer
If x=a, y=b, z=c is a solution of the system of linear equations x +8y...
Make three equations in a b and c and solve for its value..
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If x=a, y=b, z=c is a solution of the system of linear equations x +8y +7z=0 9x +2y +3z=0 x +y +z=0 such that the point(a,b,c) lies on plane x +2y +z=6 then 2a +b +c equals to a.-1 b.0 c.1 d.2?
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If x=a, y=b, z=c is a solution of the system of linear equations x +8y +7z=0 9x +2y +3z=0 x +y +z=0 such that the point(a,b,c) lies on plane x +2y +z=6 then 2a +b +c equals to a.-1 b.0 c.1 d.2? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about If x=a, y=b, z=c is a solution of the system of linear equations x +8y +7z=0 9x +2y +3z=0 x +y +z=0 such that the point(a,b,c) lies on plane x +2y +z=6 then 2a +b +c equals to a.-1 b.0 c.1 d.2? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If x=a, y=b, z=c is a solution of the system of linear equations x +8y +7z=0 9x +2y +3z=0 x +y +z=0 such that the point(a,b,c) lies on plane x +2y +z=6 then 2a +b +c equals to a.-1 b.0 c.1 d.2?.
If x=a, y=b, z=c is a solution of the system of linear equations x +8y +7z=0 9x +2y +3z=0 x +y +z=0 such that the point(a,b,c) lies on plane x +2y +z=6 then 2a +b +c equals to a.-1 b.0 c.1 d.2? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about If x=a, y=b, z=c is a solution of the system of linear equations x +8y +7z=0 9x +2y +3z=0 x +y +z=0 such that the point(a,b,c) lies on plane x +2y +z=6 then 2a +b +c equals to a.-1 b.0 c.1 d.2? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If x=a, y=b, z=c is a solution of the system of linear equations x +8y +7z=0 9x +2y +3z=0 x +y +z=0 such that the point(a,b,c) lies on plane x +2y +z=6 then 2a +b +c equals to a.-1 b.0 c.1 d.2?.
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