\begin{aligned}x+4y+3z+t&=0
\\7x-5z+7t&=4\end{aligned}

You have a system of linear equations with four variables:

\[
\begin{aligned}
1. & \quad x + 4y + 3z + t = 0 \\
2. & \quad 7x - 5z + 7t = 4
\end{aligned}
\]

Since there are four variables and only two equations, the system is underdetermined, meaning there are infinitely many solutions. The solution will depend on the values of the free variables, which we can choose arbitrarily.

### Steps to find the general solution:
1. **Express some variables in terms of others**: We'll express \(x\) and \(y\) in terms of the free variables \(z\) and \(t\).

2. **Substitute and solve**:
   - From the second equation, solve for \(x\):
     \[
     7x = 5z - 7t + 4 \quad \Rightarrow \quad x = \frac{5z - 7t + 4}{7}
     \]
   - Substitute \(x = \frac{5z - 7t + 4}{7}\) into the first equation:
     \[
     \frac{5z - 7t + 4}{7} + 4y + 3z + t = 0
     \]
   - Multiply the entire equation by 7 to eliminate the fraction:
     \[
     5z - 7t + 4 + 28y + 21z + 7t = 0
     \]
     Simplify this equation:
     \[
     28y + 26z + 4 = 0 \quad \Rightarrow \quad y = -\frac{13z + 2}{14}
     \]

### General Solution:
- \(x = \frac{5z - 7t + 4}{7}\)
- \(y = -\frac{13z + 2}{14}\)
- \(z\) and \(t\) are free variables.

Thus, the solution set is:

\[
\left( \frac{5z - 7t + 4}{7}, -\frac{13z + 2}{14}, z, t \right)
\]

This represents the general solution to the system of equations.

Would you like more details on this solution or any specific part of it? Here are some related questions:

1. How do you determine the number of solutions in a system of linear equations?
2. What are free variables, and how do they influence the solution set?
3. How can you find a particular solution from the general solution?
4. How can this solution be visualized geometrically?
5. How do systems of equations differ when they are consistent vs. inconsistent?

**Tip:** When dealing with systems of linear equations, always check if the system is underdetermined, overdetermined, or exactly determined to understand the nature of the solution set.

Learn how to solve a system of linear equations with four variables using a step-by-step approach. Understand the concept of underdetermined systems and how free variables influence the solution set. Suitable for advanced high school students studying linear algebra.

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