Question: how to slove help me
1. **Stating the problem:** You want to learn how to solve a system of equations with 3 variables.
2. **General form:** A system of 3 variables usually looks like:
$$\begin{cases} a_1x + b_1y + c_1z = d_1 \\ a_2x + b_2y + c_2z = d_2 \\ a_3x + b_3y + c_3z = d_3 \end{cases}$$
3. **Methods to solve:** The common methods are substitution, elimination, and matrix methods (like Gaussian elimination).
4. **Example:** Suppose we have:
$$\begin{cases} 2x + y - z = 8 \\ -3x - y + 2z = -11 \\ -2x + y + 2z = -3 \end{cases}$$
5. **Step 1: Choose one equation and solve for one variable.**
From the first equation:
$$y = 8 - 2x + z$$
6. **Step 2: Substitute $y$ into the other two equations:**
$$-3x - (8 - 2x + z) + 2z = -11$$
$$-2x + (8 - 2x + z) + 2z = -3$$
7. **Step 3: Simplify both:**
$$-3x - 8 + 2x - z + 2z = -11 \Rightarrow -x + z - 8 = -11$$
$$-2x + 8 - 2x + z + 2z = -3 \Rightarrow -4x + 3z + 8 = -3$$
8. **Step 4: Rearrange:**
$$-x + z = -3$$
$$-4x + 3z = -11$$
9. **Step 5: Solve this 2-variable system:**
Multiply the first by 3:
$$-3x + 3z = -9$$
Subtract from second:
$$(-4x + 3z) - (-3x + 3z) = -11 - (-9)$$
$$-4x + 3z + 3x - 3z = -11 + 9$$
$$-x = -2 \Rightarrow x = 2$$
10. **Step 6: Substitute $x=2$ back:**
$$-x + z = -3 \Rightarrow -2 + z = -3 \Rightarrow z = -1$$
11. **Step 7: Substitute $x=2$, $z=-1$ into $y = 8 - 2x + z$:**
$$y = 8 - 2(2) + (-1) = 8 - 4 - 1 = 3$$
12. **Final answer:**
$$x=2, y=3, z=-1$$
This is the solution to the system.
Remember, the key steps are isolating one variable, substituting, simplifying, and solving step-by-step.