You have a system of linear equations:

\[
4x + y - 2z + t = 1 
\]
\[
2x + y + 3z - 2t = 3
\]

To solve this system, we can use methods like substitution, elimination, or matrix techniques. Since we have two equations with four variables, the system is underdetermined, meaning it can have infinitely many solutions depending on the values of two of the variables. Here’s the general approach to find the solution set:

### Step 1: Express one variable in terms of others.
We can solve one of the equations for one variable and then substitute into the other equation.

Let’s solve the first equation for \( y \):

\[
y = 1 - 4x + 2z - t
\]

### Step 2: Substitute into the second equation.
Substituting \( y = 1 - 4x + 2z - t \) into the second equation:

\[
2x + (1 - 4x + 2z - t) + 3z - 2t = 3
\]

Simplify:

\[
2x + 1 - 4x + 2z - t + 3z - 2t = 3
\]

\[
-2x + 5z - 3t = 2
\]

### Step 3: Write the solution in parametric form.
Now we have:

\[
y = 1 - 4x + 2z - t
\]
\[
-2x + 5z - 3t = 2
\]

Let’s solve the second equation for \( x \):

\[
x = \frac{5z - 3t - 2}{2}
\]

Now, we express \( y \) in terms of \( z \) and \( t \):

\[
y = 1 - 4\left(\frac{5z - 3t - 2}{2}\right) + 2z - t
\]

Simplify this to express \( y \) explicitly in terms of \( z \) and \( t \).

### Final Parametric Solution:
The solution can be written as:

\[
x = \frac{5z - 3t - 2}{2}, \quad y = 1 - 10z + 7t, \quad z = z, \quad t = t
\]

Where \( z \) and \( t \) are free parameters, meaning they can take any real number values.

### Summary:
The solution to the system of equations is given by:

\[
(x, y, z, t) = \left(\frac{5z - 3t - 2}{2}, 1 - 10z + 7t, z, t\right)
\]

This represents an infinite set of solutions, where \( z \) and \( t \) can be any real numbers.

---

Do you want more details on any of these steps or have any questions?

### Related Questions:
1. How do you determine if a system of linear equations is underdetermined?
2. What is the geometric interpretation of the solution to this system?
3. How can matrix methods be used to solve this system?
4. What is the role of free parameters in the solution of a linear system?
5. How does the solution change if more equations are added to the system?

### Tip:
When dealing with underdetermined systems, it's helpful to think of the solutions as lying on a plane, line, or higher-dimensional space, depending on the number of free parameters.

Learn how to solve the system of linear equations 4x + y - 2z + t = 1 and 2x + y + 3z - 2t = 3 with a detailed step-by-step solution. This problem covers key algebraic concepts including systems of linear equations and parametric solutions. Suitable for high school students.

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