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3 years ago

I can travel 250 miles in 90 minutes, how far can i get in 6 hours

Mathematics

1 answer:

german3 years ago

4 0

90 minutes = 1.5 hours.

Divide total miles driven by the time to find how many miles are driven in 1 hour:

250 miles / 1.5 hours = 166 2/3 miles per hour.

Multiply the miles driven in one hour by 6 hours:

166 2/3 x 6 = 1000 miles

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One of the obvious non-trivial solutions is $(x_1, x_2, x_3)=(-2, 3, -5)$ .

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$A=\left[\begin{array}{ccc}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&3_{23}\\a_{31}&a_{32}&a_{33}\end{array}\right]$

The observed system $Ax=0$ after multiplying looks like this

$Ax=0 \iff \left[\begin{array}{ccc}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{array}\right] \cdot \left[\begin{array}{ccc}x_1\\x_2\\x_3\end{array}\right] =0 \iff \\ \\a_{11}x_1+a_{12}x_2+a_{13}x_3=0\\a_{21}x_1+a_{22}x_2+a_{23}x_3=0\\a_{31}x_1+a_{32}x_2+a_{33}x_3=0\\\\$

Since we now that $-2A_1+3A_2-5A_3=0$ , where $A_i\ ,\ i=1, 2, 3$ are the columns of the matrix A, we actually know this:

$-2\cdot \left[\begin{array}{ccc}a_{11}\\a_{21}\\a_{31}\end{array}\right] +3\cdot \left[\begin{array}{ccc}a_{12}\\a_{22}\\a_{32}\end{array}\right] -5\cdot \left[\begin{array}{ccc}a_{13}\\a_{23}\\a_{33}\end{array}\right] =\left[\begin{array}{ccc}0\\0\\0\end{array}\right]$

Once we multiply and sum up these 3 by 1 matrices, we get that these equations hold:

$-2a_{11}+3a_{12}-5a_{13}=0\\-2a_{21}+3a_{22}-5a_{23}=0\\-2a_{31}+3a_{32}-5a_{33}=0$

This actually means that the solution to the previously observed system of equations (or equivalently, our system $Ax=0$ ) has a non-trivial solution $(x_1, x_2, x_3)=(-2, 3, -5)$ .

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