To solve the system of equations using elimination, follow these steps:

Given equations:

1. $y = -3x + 6$ (Equation 1)
2. $2x - 3y = 4$ (Equation 2)

Step 1: Substitute $y$ from Equation 1 into Equation 2

From Equation 1, $y = -3x + 6$. Substitute this into Equation 2: $2x - 3(-3x + 6) = 4$

Simplify: $2x + 9x - 18 = 4$

Combine like terms: $11x - 18 = 4$

Step 2: Solve for $x$

$11x = 4 + 18$ $11x = 22$ $x = 2$

Step 3: Solve for $y$

Substitute $x = 2$ into Equation 1: $y = -3(2) + 6$ $y = -6 + 6$ $y = 0$

Final Solution:

The solution to the system of equations is: $x = 2, \; y = 0$

Verification:

1. Substitute $x = 2$, $y = 0$ into Equation 1: $y = -3(2) + 6 = 0 \; \text{(Correct)}.$
2. Substitute $x = 2$, $y = 0$ into Equation 2: $2(2) - 3(0) = 4 \; \text{(Correct)}.$

The solution satisfies both equations.

---

Would you like a detailed breakdown of any step?

Related Questions:

1. How does the substitution method differ from the elimination method?
2. What are some scenarios where elimination is more efficient than substitution?
3. How can systems of equations be solved graphically, and how does the solution compare?
4. Can we confirm the solution using matrix methods like Gaussian elimination?
5. What are the conditions for a system of equations to have no solution or infinitely many solutions?

Tip:

When using elimination or substitution, always simplify equations as much as possible to avoid errors in algebraic manipulation.