Determine whether the given (2 × 3) system of linear equations represents coincident planes (that is, the same plane), two parallel planes, or two planes whose intersection is a line. In the latter case, give the parametric equations for the line, that is, give equations of the form x=at + b, y=ct + d, z=et + f. x1 + 2x2 − x3=2 x1 + x2 + x3=3

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Determine whether the given (2 × 3) system of linear equations represents coincident planes (that is, the same plane), two parallel planes, or two planes whose intersection is a line. In the latter case, give the parametric equations for the line, that is, give equations of the form  x=at + b, y=ct + d, z=et + f.  x1 + 2x2 − x3=2  x1 + x2 + x3=3

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Step 1  The given system of linear equations  x1 + 2x2 − x3=2  x1 + x2 + x3=3  We need to determine whether the given system of linear equations represents coicident planes, two parallel planes, or two planes whose intersection is a line.  Let n1=(1, 2, −1) and n2=(1, 1, 1,) be normal vectors of both the equations.  Then n1 and n2 are not parallel.  Let x3=t then replace x3=t in the system of linear equations  x1 + 2x2 − t=2(1)  x1 + x2 + t=3(2)  (1)⇒ x1=2 − 2x2 + t  Replace x1=2 − 2x2 + t in the second equation  2 − 2x2 + t + x2 + t=3  −x2=1 − 2t  x2=2t − 1  Replace x2=2t − 1 and x3=t ∈ x1=2 − 2x2 + t, then  x1=2 − 2(2t − 1) + t  =2 − 4t + 2 − t  =−3t + 4  Hence, the plane intersect in a line and the parametric equations are  x1= −3t + 4, x2=2t − 1 and x3=t

Determine whether the given (2 times 3) system of linear equations represents coincident planes (that is, the same plane), two parallel planes, or two

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