Tyra

2021-02-03

Determine whether the given $(2\text{}\times \text{}3)$ 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

$x=at\text{}+\text{}b,\text{}y=ct\text{}+\text{}d,\text{}z=et\text{}+\text{}f.$

${x}_{1}\text{}+\text{}2{x}_{2}\text{}-\text{}{x}_{3}=2$

${x}_{1}\text{}+\text{}{x}_{2}\text{}+\text{}{x}_{3}=3$

i1ziZ

Skilled2021-02-04Added 92 answers

Step 1

The given system of linear equations

${x}_{1}\text{}+\text{}2{x}_{2}\text{}-\text{}{x}_{3}=2$

${x}_{1}\text{}+\text{}{x}_{2}\text{}+\text{}{x}_{3}=3$

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${n}_{1}=(1,\text{}2,\text{}-1)\text{}{\textstyle \phantom{\rule{1em}{0ex}}}\text{and}{\textstyle \phantom{\rule{1em}{0ex}}}\text{}{n}_{2}=(1,\text{}1,\text{}1,)$ be normal vectors of both the equations.

Then$n}_{1}\text{}{\textstyle \phantom{\rule{1em}{0ex}}}\text{and}{\textstyle \phantom{\rule{1em}{0ex}}}\text{}{n}_{2$ are not parallel.

Let${x}_{3}=t\text{}then\text{}replace\text{}{x}_{3}=t$ in the system of linear equations

${x}_{1}\text{}+\text{}2{x}_{2}\text{}-\text{}t=2$ (1)

${x}_{1}\text{}+\text{}{x}_{2}\text{}+\text{}t=3$ (2)

(1)$\Rightarrow \text{}{x}_{1}=2\text{}-\text{}2{x}_{2}\text{}+\text{}t$

Replace${x}_{1}=2\text{}-\text{}2{x}_{2}\text{}+\text{}t$ in the second equation

$2\text{}-\text{}2{x}_{2}\text{}+\text{}t\text{}+\text{}{x}_{2}\text{}+\text{}t=3$

$-{x}_{2}=1\text{}-\text{}2t$

${x}_{2}=2t\text{}-\text{}1$

Replace${x}_{2}=2t\text{}-\text{}1\text{}{\textstyle \phantom{\rule{1em}{0ex}}}\text{and}{\textstyle \phantom{\rule{1em}{0ex}}}\text{}{x}_{3}=t\text{}\in \text{}{x}_{1}=2\text{}-\text{}2{x}_{2}\text{}+\text{}t$ , then

${x}_{1}=2\text{}-\text{}2(2t\text{}-\text{}1)\text{}+\text{}t$

$=2\text{}-\text{}4t\text{}+\text{}2\text{}-\text{}t$

$=-3t\text{}+\text{}4$

Hence, the plane intersect in a line and the parametric equations are

${x}_{1}=\text{}-3t\text{}+\text{}4,\text{}{x}_{2}=2t\text{}-\text{}1\text{}{\textstyle \phantom{\rule{1em}{0ex}}}\text{and}{\textstyle \phantom{\rule{1em}{0ex}}}\text{}{x}_{3}=t$

The given system of linear equations

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

Then

Let

(1)

Replace

Replace

Hence, the plane intersect in a line and the parametric equations are

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