To balance the chemical equation \( \text{Cu} + X \text{HNO}_3 \rightarrow \text{Cu(NO}_3)_2 + Y \text{NO}_2 + Z \text{H}_2\text{O} \), we will follow a systematic approach to ensure that the number of atoms of each element is the same on both sides of the equation.
### Step-by-Step Solution:
1. **Identify the Elements**:
The elements involved in the reaction are Copper (Cu), Nitrogen (N), Oxygen (O), and Hydrogen (H).
2. **Write Down the Unbalanced Equation**:
\[
\text{Cu} + X \text{HNO}_3 \rightarrow \text{Cu(NO}_3)_2 + Y \text{NO}_2 + Z \text{H}_2\text{O}
\]
3. **Balance Copper (Cu)**:
On the reactant side, we have 1 Cu atom, and on the product side, we also have 1 Cu atom in \(\text{Cu(NO}_3)_2\). Therefore, copper is already balanced.
4. **Balance Nitrogen (N)**:
In the product side, \(\text{Cu(NO}_3)_2\) contains 2 nitrogen atoms, and \(Y \text{NO}_2\) contains \(Y\) nitrogen atoms. Therefore, the total nitrogen on the product side is \(2 + Y\).
On the reactant side, there are \(X\) nitrogen atoms from \(X \text{HNO}_3\).
Thus, we set up the equation:
\[
X = 2 + Y
\]
5. **Balance Oxygen (O)**:
In \(\text{Cu(NO}_3)_2\), there are 6 oxygen atoms (from 2 nitrate ions), and in \(Y \text{NO}_2\), there are \(2Y\) oxygen atoms. In \(Z \text{H}_2\text{O}\), there are \(Z\) oxygen atoms.
Therefore, the total oxygen on the product side is:
\[
6 + 2Y + Z
\]
On the reactant side, there are \(3X\) oxygen atoms from \(X \text{HNO}_3\).
Thus, we set up the equation:
\[
3X = 6 + 2Y + Z
\]
6. **Balance Hydrogen (H)**:
On the product side, there are \(2Z\) hydrogen atoms from \(Z \text{H}_2\text{O}\).
On the reactant side, there are \(X\) hydrogen atoms from \(X \text{HNO}_3\).
Thus, we set up the equation:
\[
X = 2Z
\]
7. **Solve the Equations**:
Now we have a system of equations:
1. \(X = 2 + Y\)
2. \(3X = 6 + 2Y + Z\)
3. \(X = 2Z\)
From equation 1, we can express \(Y\) in terms of \(X\):
\[
Y = X - 2
\]
Substitute \(Y\) into equation 2:
\[
3X = 6 + 2(X - 2) + Z
\]
Simplifying gives:
\[
3X = 6 + 2X - 4 + Z \implies 3X = 2X + 2 + Z \implies X - 2 = Z
\]
Now substitute \(Z\) into equation 3:
\[
X = 2(X - 2) \implies X = 2X - 4 \implies X = 4
\]
Now substitute \(X\) back to find \(Y\) and \(Z\):
\[
Y = 4 - 2 = 2
\]
\[
Z = 4 - 2 = 2
\]
8. **Final Values**:
Thus, the values are:
\[
X = 4, \quad Y = 2, \quad Z = 2
\]
### Conclusion:
The balanced equation is:
\[
\text{Cu} + 4 \text{HNO}_3 \rightarrow \text{Cu(NO}_3)_2 + 2 \text{NO}_2 + 2 \text{H}_2\text{O}
\]
