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Radda [10]
3 years ago
12

Determine all possible digit replacements for x so that the first number is divisible by the second.

Mathematics
1 answer:
svet-max [94.6K]3 years ago
4 0

Answer:

Step-by-step explanation:

x must be an even number from 0 to 9. So x can be

0 2 4 6 8

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Find a fundamental matrix solution for the system x 0 1 7x1 + 4x2 + 12x3, x 0 2 x1 + 2x2 + x3, x 0 3 −3x1 − 2x2 − 5x3. Then find
liq [111]

Here is the correct format for the question.

x'_1 = 7x_1 +4x_2+ 12x_3 , \ \ x'_2 = x_1 + 2x_2 + x_3 ,  \ \ x'_3 = -3x_1 -2x_2 -5x_3 . Then find the solution that satisfies  x \limits ^{\to} = \left[\begin{array}{c}0\\1\\-2\end{array}\right]

Answer:

\mathbf{c_1 = -3, c_2 = 2, c_3 = 2}

Step-by-step explanation:

From the figures given above:

the matrix can be computed as,

\left[\begin{array}{ccc}7&4&12\\1&2&1\\-3&-2&-5\end{array}\right] x \limits ^{\to} = \left[\begin{array}{c}x_1\\x_2\\x_3\end{array}\right] = x \limits ^{\to} = \left[\begin{array}{c}x_1'\\x_2'\\x_3'\end{array}\right]

The first thing we need to carry out is to determine the eigenvalues of A,

where:

A = \left[\begin{array}{ccc}7&4&12\\1&2&1\\-3&-2&-5\end{array}\right]

|A-rI|=0

\begin {vmatrix} 7-r&4&12\\1&2-r&1\\-3&-2&-5-r  \end {vmatrix}=0

the eigenvalues are r = 0, 1, 3

However, the eigenvector correlated to each eigenvalue can be calculated as follows.

suppose  r = 0

(A - rI) x = 0

\left[\begin{array}{ccc}7&4&12\\1&2&1\\-3&-2&-5\end{array}\right] \left[\begin{array}{c}x_1\\x_2\\x_3\end{array}\right] = \left[\begin{array}{c}0\\0\\0\end{array}\right]

now the eigenvector is \left[\begin{array}{c}-4\\1\\2\end{array}\right]

However, for eigenvalue = 1, we have :  \left[\begin{array}{c}-4\\1\\2\end{array}\right]

for eigenvalue = 3, we have:\left[\begin{array}{c}-2\\-1\\1\end{array}\right]

The solution now can be computed as :

x(t)=  c_1 \left[\begin{array}{c}-4\\1\\2\end{array}\right] + c_2e^t \left[\begin{array}{c}-4\\3\\1\end{array}\right]+ c_3e^{3t} \left[\begin{array}{c}-2\\-1\\1\end{array}\right]

Similarly, the fundamental matrix solution is:

\left[\begin{array}{ccc}-4&-4e^t&-2e^{3t}\\1&3e^t&-e^{3t}\\2&e^t&e^{3t}\end{array}\right]

-4c_1 -4c_2-2c_3 =0 \\ \\ c_1 + 3c_2 -c_3 = 1\\ \\ 2c_1+c_2 +c_3 = -2

Solving the above equation, we get:

\mathbf{c_1 = -3, c_2 = 2, c_3 = 2}

5 0
3 years ago
A number of students are evenly spaced around the circle. The fourth student is directly opposite the tenth student. How many st
natta225 [31]

Answer:

  • 12 students

Step-by-step explanation:

<u>Between the fourth and tenth there are:</u>

  • 9 - 4 = 5 students

<u>The total number is:</u>

  • 2 + 5*2 = 12 students
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Step-by-step explanation:

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