1) Two positive integers differ by \(5686\). If the digit \(3\) is rejected from the right of the first number and the digit \(4\) is annexed to the right of the second number, then the newly obtained numbers will be equal to each other. Find the two original numbers.
2) Find all the ordered pairs \(\left(x,y\right)\) that solve this system. \(\begin{cases} 5(x^2 -36) + \frac{2}{y}=11\\ x^2-36+\frac{1}{y}=4 \end{cases}\)
3) Solve the system: \(\begin{cases} 2x+3y-z=-1 \\ -x+5y+3z=-10 \\ 3x-y-6z=5\end{cases}\)
4) Find the value of \(k\) so that the following system has a unique solution. \(\begin{cases} 2x+5y=9 \\ x+2y=4 \\ kx+6y=7 \end{cases}\)
5) A store began selling its line of fall sweaters. On the first day, \(6\) cotton, \(4\) acrylic, and \(5\) wool sweaters were sold. On the second day, \(3\) cotton, \(5\) acrylic, and \(8\) wool sweaters were sold. On the third day, \(4\) cotton, \(1\) acrylic, and \(4\) wool sweaters were sold. Total sales for the three days were \(\$395\), \(\$505\), and \(\$260\), respectively. What was the sale price for each sweater?
6) Find the value of \(k\) so that the following system is inconsistent. \(\begin{cases} \frac{1}{3}x+4y=7 \\ kx+7y=5 \end{cases}\)
7) Solve the system: \(\begin{cases}\large\frac{2}{x}-\frac{1}{y}+\frac{6}{z}=0\\\large\frac{4}{x}+\frac{1}{y}-\frac{3}{z}=-14\\\large\frac{2}{x}+\frac{2}{y}+\frac{3}{z}=14\end{cases}\)
8) Solve the system: \(\begin{cases}\large\frac{1}{x}+\frac{1}{y}=\frac{2}{3}\\\large\frac{1}{y}+\frac{1}{z}=-\frac{1}{6}\\\large\frac{1}{x}+\frac{1}{z}=\frac{1}{6}\end{cases}\)
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2) Find all the ordered pairs \(\left(x,y\right)\) that solve this system. \(\begin{cases} 5(x^2 -36) + \frac{2}{y}=11\\ x^2-36+\frac{1}{y}=4 \end{cases}\)
3) Solve the system: \(\begin{cases} 2x+3y-z=-1 \\ -x+5y+3z=-10 \\ 3x-y-6z=5\end{cases}\)
4) Find the value of \(k\) so that the following system has a unique solution. \(\begin{cases} 2x+5y=9 \\ x+2y=4 \\ kx+6y=7 \end{cases}\)
5) A store began selling its line of fall sweaters. On the first day, \(6\) cotton, \(4\) acrylic, and \(5\) wool sweaters were sold. On the second day, \(3\) cotton, \(5\) acrylic, and \(8\) wool sweaters were sold. On the third day, \(4\) cotton, \(1\) acrylic, and \(4\) wool sweaters were sold. Total sales for the three days were \(\$395\), \(\$505\), and \(\$260\), respectively. What was the sale price for each sweater?
6) Find the value of \(k\) so that the following system is inconsistent. \(\begin{cases} \frac{1}{3}x+4y=7 \\ kx+7y=5 \end{cases}\)
7) Solve the system: \(\begin{cases}\large\frac{2}{x}-\frac{1}{y}+\frac{6}{z}=0\\\large\frac{4}{x}+\frac{1}{y}-\frac{3}{z}=-14\\\large\frac{2}{x}+\frac{2}{y}+\frac{3}{z}=14\end{cases}\)
8) Solve the system: \(\begin{cases}\large\frac{1}{x}+\frac{1}{y}=\frac{2}{3}\\\large\frac{1}{y}+\frac{1}{z}=-\frac{1}{6}\\\large\frac{1}{x}+\frac{1}{z}=\frac{1}{6}\end{cases}\)
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