https://file.aiquickdraw.com/mathbot/file/1729078461484lb8f2a79.jpg

The system of linear equations from the uploaded image is:

\[
\begin{aligned}
1) & \quad 2x - 4y + z = -10 \\
2) & \quad 3x + y + 2z = 60 \\
3) & \quad -2x + y - z = -20
\end{aligned}
\]

We will solve this system step-by-step using **substitution or elimination**.

---

### Step 1: Simplify where possible

From Equation (3):
\[
-2x + y - z = -20 \implies y = 2x + z - 20 \quad \text{(Rearrange for \(y\))}
\]
We'll substitute this expression for \(y\) in the other two equations.

---

### Step 2: Substitute \(y = 2x + z - 20\) into the first two equations

**Substitute into Equation (1):**

\[
2x - 4(2x + z - 20) + z = -10
\]

Simplify:

\[
2x - 8x - 4z + 80 + z = -10
\]
\[
-6x - 3z = -90 \implies 2x + z = 30 \quad \text{(Divide by \(-3\))}
\]

So, Equation (1) simplifies to:

\[
2x + z = 30 \quad \text{(Let's call this Eq (4))}
\]

**Substitute into Equation (2):**

\[
3x + (2x + z - 20) + 2z = 60
\]

Simplify:

\[
3x + 2x + z - 20 + 2z = 60
\]
\[
5x + 3z = 80 \quad \text{(Let's call this Eq (5))}
\]

---

### Step 3: Solve the two new equations

We now have:
1. \(2x + z = 30\) \quad (Eq (4))
2. \(5x + 3z = 80\) \quad (Eq (5))

**Solve Eq (4) for \(z\):**

\[
z = 30 - 2x
\]

**Substitute into Eq (5):**

\[
5x + 3(30 - 2x) = 80
\]

Simplify:

\[
5x + 90 - 6x = 80
\]
\[
-x = -10 \implies x = 10
\]

---

### Step 4: Find \(z\) and \(y\)

Using \(x = 10\) in Eq (4):

\[
2(10) + z = 30 \implies 20 + z = 30 \implies z = 10
\]

Now, substitute \(x = 10\) and \(z = 10\) into the expression for \(y\):

\[
y = 2(10) + 10 - 20 = 20 + 10 - 20 = 10
\]

---

### Step 5: Solution

The solution to the system of equations is:

\[
x = 10, \quad y = 10, \quad z = 10
\]

---

Do you need further details or have any questions?

---

Here are 5 related questions to extend your understanding:
1. What are the different methods to solve a system of linear equations?
2. How can we determine if a system of equations has no solution, one solution, or infinitely many solutions?
3. What is the determinant of a system, and how does it relate to the existence of solutions?
4. Can you solve the system using matrix operations such as Gaussian elimination?
5. How would the solution change if one of the equations was altered slightly?

**Tip:** Substitution works best when one equation is already solved for a variable, while elimination is useful when you need to eliminate terms quickly.

Tentukan himpunan penyelesaian dari persamaan linear berikut: 2x - 4y + z = -10, 3x + y + 2z = 60, -2x + y - z = 20.

Learn how to solve a system of three linear equations step-by-step: 2x - 4y + z = -10, 3x + y + 2z = 60, and -2x + y - z = 20. This problem involves using substitution and elimination methods, suitable for students in Grades 10-12.

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