Conditions for a system of linear equations to have infinite solutions
If x=a, y=b...
Question
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 the plane x+2y+z=6, then 2a+b+c equals.
A
0
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B
2
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C
1
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D
−1
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Solution
The correct option is C1 x+8y+7z=0 ........(1) 9x+2y+3z=0 .........(2) x+y+z=0 ........(3) Multiply eqn. (3) by 3 & subtract from eqn. (2) 9x+2y+3z=0 3x±3y±3z=0 _____________________ 6x−y=0 _____________________ y=6x .........(4) Substitute in Eqn (1). we get x+8(6x)+7z=0 49x+7z=0 7x+z=0 z=−7x .........(5) ⇒x=a, y=b.z=c is the solutions of the system of linear equation ∴b=6a and c=−7a .......(6) Also, (a, b, c) lies on the plane x+2y+z=6 ∴a+2b+c=6 ........(7) Substitute eq. (6) in eq(7) ∴a+2(6a)−7a=6 ∴a+12a−7a=6 ∴6a=6 ∴a=1 So, b=6 and c=−7 ∴2a+b+c=2(1)+6−7=1.