solve this system of equations.  3x+2y-z=7, and x-4y+2x=0

To solve the system of equations, we need to rewrite and solve them simultaneously. The given system of equations is:

1. \(3x + 2y - z = 7\)
2. \(x - 4y + 2x = 0\)

First, let's simplify the second equation. Combining like terms, we get:

\[3x - 4y = 0\]

Now we have the system:

1. \(3x + 2y - z = 7\)
2. \(3x - 4y = 0\)

Next, let's solve this system step-by-step.

**Step 1:** Solve the second equation for \(x\):

\[3x - 4y = 0\]
\[3x = 4y\]
\[x = \frac{4y}{3}\]

**Step 2:** Substitute \(x = \frac{4y}{3}\) into the first equation:

\[3\left(\frac{4y}{3}\right) + 2y - z = 7\]

Simplify:

\[4y + 2y - z = 7\]
\[6y - z = 7\]

**Step 3:** Solve for \(z\) in terms of \(y\):

\[z = 6y - 7\]

**Step 4:** Substitute back \(x = \frac{4y}{3}\) and \(z = 6y - 7\) to find the values of \(x\), \(y\), and \(z\).

Since we have two equations and three variables, \(y\) can take any real number. Let's choose a specific value for \(y\) to find a particular solution. For simplicity, let \(y = 1\):

\[x = \frac{4(1)}{3} = \frac{4}{3}\]
\[z = 6(1) - 7 = -1\]

So, a particular solution to the system of equations is:

\[x = \frac{4}{3}, y = 1, z = -1\]

**General Solution:**

The general solution will depend on the parameter \(y\). Hence, the solutions can be written as:

\[x = \frac{4y}{3}, z = 6y - 7\]

Would you like more details or have any questions?

Here are 8 related questions to expand your understanding:

1. How can you solve the system of equations using matrix methods?
2. What is the significance of having more variables than equations in a system?
3. Can you graph the system of equations and interpret the solution?
4. How would you solve the system using substitution method in a different way?
5. How can the elimination method be applied to this system of equations?
6. What are the implications of a system of equations having infinitely many solutions?
7. How does the solution change if we change the value of one of the constants in the equations?
8. What are the differences between consistent, inconsistent, and dependent systems of equations?

**Tip:** When dealing with systems of linear equations, always check if the system is underdetermined, overdetermined, or exactly determined to choose the most appropriate solution method.

Learn how to solve the system of equations 3x + 2y - z = 7 and x - 4y + 2x = 0 step-by-step with detailed solutions and examples. This problem involves linear equations and simultaneous equations, suitable for high school students.

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