https://r2.erweima.ai/mathbot/image/17257685592356vuhzhqy.jpg

The system of linear equations in the image is:

\[
\begin{aligned}
3x + 2y + z &= 13 \\
4x &= 7 + 5y
\end{aligned}
\]

To solve this, let's rewrite the second equation:

\[
4x - 5y = 7
\]

Now the system becomes:

1. \( 3x + 2y + z = 13 \)
2. \( 4x - 5y = 7 \)

We have two equations with three variables. This indicates that there might be infinitely many solutions depending on how we express one variable in terms of the others. Let's solve for \(x\) and \(y\) first, and then express \(z\) based on them.

First, solve for \(x\) and \(y\).The solution for \(x\) and \(y\) is:

\[
x = \frac{79}{23}, \quad y = \frac{31}{23}
\]

Now, let's find the value of \(z\) by substituting these values into the first equation \(3x + 2y + z = 13\).The value of \(z\) is \(0\).

Thus, the solution to the system of equations is:

\[
x = \frac{79}{23}, \quad y = \frac{31}{23}, \quad z = 0
\]

Would you like a detailed explanation of how each step was done?

Here are 5 related questions you might find useful:
1. How can we verify the solution to this system of equations?
2. What is the geometric interpretation of a system with two equations and three unknowns?
3. Can the system be solved using matrices or determinants?
4. How would the solution change if the system had no solutions or infinitely many solutions?
5. What techniques are used to solve more complex nonlinear systems of equations?

**Tip:** Always check your solution by plugging the values back into the original system of equations to ensure correctness.

Resuelve el siguiente sistema de ecuaciones lineales: 3x + 2y + z = 13, 4x = 7 + 5y

Learn how to solve the system of linear equations: 3x + 2y + z = 13 and 4x = 7 + 5y. This solution involves key algebraic concepts like the substitution method and solving systems with three variables. Suitable for students in Grades 10-12.

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