Determine whether the ordered pair is a solution to the given system of linear equations.

(5,3)

vestirme4
2021-03-12
Answered

Determine whether the ordered pair is a solution to the given system of linear equations.

(5,3)

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lamusesamuset

Answered 2021-03-13
Author has **93** answers

Given

The given ordered pair for this system is (5, 3).

The ordered pair is the solution of the system it must be satisfy both the equations.

Step 1

Substitute

Substitute

Since the ordered pair satisfies the system of linear equations.

Hence, the ordered pair (5, 3) is a solution to the given system of linear equations.

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Solve the following system of linear equations in terms of parameter $a\in \mathbb{R}$ and explain geometric interpretation of this system:

$ax+y+z=1,2x+2ay+2z=3,x+y+az=1$.

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$\left[\begin{array}{cccc}a& 1& 1& 1\\ 2& 2a& 2& 3\\ 1& 1& a& 1\end{array}\right]$

Row echelon form of this matrix is

$\left[\begin{array}{cccc}1& 1& a& 1\\ 0& 2(a-1)& 2(1-a)& 1\\ 0& 0& (1-a)(a+2)& (3-2a)/2\end{array}\right]$

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Question: Is this geometric interpretation correct?

$ax+y+z=1,2x+2ay+2z=3,x+y+az=1$.

By Cronecker Capelli's theorem, we get:

$\left[\begin{array}{cccc}a& 1& 1& 1\\ 2& 2a& 2& 3\\ 1& 1& a& 1\end{array}\right]$

Row echelon form of this matrix is

$\left[\begin{array}{cccc}1& 1& a& 1\\ 0& 2(a-1)& 2(1-a)& 1\\ 0& 0& (1-a)(a+2)& (3-2a)/2\end{array}\right]$

System is inconsistent for $a=1\vee a=-2$. For every other value of $a$, system has unique solution.

For every $a$ instead of $a=1\wedge a=-2$ there are three planes that intersect at a point.

Question: Is this geometric interpretation correct?

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