Problem 1: Xavier, Yuri and Zyra are artists that want to paint a mural in the school. The 3 of them together take 8 hours to complete the mural. Yuri takes twice as long as Xavier to complete the mural if they were working individually. Zyra takes 10 hours less than Xavier if they were working individually. How long would each of them to paint the mural if they were working individually?

## Step 1: Setting up Variables<br />### Let $x$ be the time Xavier takes to complete the mural, $y$ be the time Yuri takes, and $z$ be the time Zyra takes, all measured in hours.<br /><br />## Step 2: Formulating Equations<br />### We are given that Yuri takes twice as long as Xavier, so $y = 2x$. Zyra takes 10 hours less than Xavier, so $z = x - 10$. Working together, they take 8 hours. The combined work rate is $\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = \frac{1}{8}$.<br /><br />## Step 3: Substituting and Solving for x<br />### Substitute the expressions for $y$ and $z$ into the combined work rate equation: $\frac{1}{x} + \frac{1}{2x} + \frac{1}{x-10} = \frac{1}{8}$. Simplify to get $\frac{3}{2x} + \frac{1}{x-10} = \frac{1}{8}$. Multiplying both sides by $8(2x)(x-10)$ gives $12(x-10) + 16x = 2x(x-10)$, which simplifies to $2x^2 - 48x + 120 = 0$. Dividing by 2 gives $x^2 - 24x + 60 = 0$.<br /><br />## Step 4: Solving the Quadratic Equation<br />### Solving the quadratic equation $x^2 - 24x + 60 = 0$ using the quadratic formula, we get $x = \frac{-(-24) \pm \sqrt{(-24)^2 - 4(1)(60)}}{2(1)} = \frac{24 \pm \sqrt{576 - 240}}{2} = \frac{24 \pm \sqrt{336}}{2} = \frac{24 \pm 4\sqrt{21}}{2} = 12 \pm 2\sqrt{21}$. Since $z = x - 10$, $x$ must be greater than 10. Therefore, $x = 12 + 2\sqrt{21} \approx 21.17$ hours.<br /><br />## Step 5: Calculating y and z<br />### Now, we can find $y$ and $z$. $y = 2x = 2(12 + 2\sqrt{21}) = 24 + 4\sqrt{21} \approx 42.33$ hours. $z = x - 10 = 12 + 2\sqrt{21} - 10 = 2 + 2\sqrt{21} \approx 11.17$ hours.