Number problems

1. Determine a number with the same result if we multiply it by ten as if we add 10 to it.
2. Věra is twice as old today as her sister Judy. In five years, Judy will be only 1/3 years younger than Věra. How old are Věra and Judy today?
3. Find the pair of integers to which it holds:
a + b = c
a * b = c

Final Answer:

x =  1.1111
V =  10
J =  5
a =  2
b =  2

Step-by-step explanation:

$$\begin{gathered} 10x=x+10 \\ 10\cdot x=x+10 \\ 9x=10 \\ x=\frac{10}{9}=1.11111111 \\ x=\frac{10}{9}\doteq 1.111111=1.1111 \end{gathered}$$

$$\begin{gathered} r=1-\frac{1}{3}=\frac{1\cdot 3}{3}-\frac{1}{3}=\frac{3}{3}-\frac{1}{3}=\frac{3-1}{3}=\frac{2}{3}\doteq 0.6667 \\ V=2J \\ J+5=r\cdot (V+5) \\ V=2\cdot J \\ J+5=0.66666666666667\cdot (V+5) \\ 2J-V=0 \\ J-0.666667V=-1.666667 \\ Row2-\frac{1}{2}\cdot Row1\to Row2 \\ -0.17V=-1.67 \\ V=\frac{-1.66666667}{-0.16666667}=10 \\ J=\frac{0+V}{2}=\frac{0+10}{2}=5 \\ J=5 \\ V=10 \end{gathered}$$

$$\begin{gathered} 2+2=2\cdot 2=4 \\ a=2 \end{gathered}$$

$b=2$

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