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Systems of Linear Equations

If one Bitcoin and 100,000 Dogecoin are worth $480, and two Bitcoin and 150,000 Dogecoin are worth $948, would you rather have one Bitcoin or 100,000 Dogecoin?

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At 6pm, 15 boys left school, and then the remaining children could be split evenly into groups each containing 2 girls and 1 boy.

At 7pm, 45 girls left school, and then the remaining children could be split evenly into groups each containing 1 girl and 5 boys.

How many boys were there before 6pm?

Given that \(x=y=0\) is not the only solution to the following system of linear equations, determine all the possible values of \(k:\) \[\begin{align} x+2y &= kx \\ 2x+y &= ky . \end{align}\]

In splitting up $1000, a group of three married couples agrees upon the following plan:

The wives receive a total of $396, of which Mary gets $10 more than Diane, and Ellen gets $10 more than Mary.

Bill Brown gets twice as much as his wife, Henry Hobson gets the same as his wife, and John Jones gets 50 percent more than his wife.

What are the full names of the three wives?

Note: Assume that the wives have the same last names as their husbands.

Today, in a 10-member committee, an old member was replaced by a young member. As such, the average age is the same today as it was 4 years ago.

What is the (positive) difference in ages between the new member and the replaced old member?

\[ \begin{eqnarray} \frac{1}{x} + \frac{1}{y} &=& \frac{1}{3}\\ \frac{1}{x} + \frac{1}{z} &=& \frac{1}{5}\\ \frac{1}{y} + \frac{1}{z} &=& \frac{1}{7} \\ \end{eqnarray} \]

Given the system of equations above, what is the value of \( \frac{z}{y}\)?

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