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

The system of equations is:

1. $2x - 3y = 8$ (Equation 1)
2. $4x - y = 6$ (Equation 2)

Step 1: Align the equations for elimination

We want to eliminate one of the variables. In this case, it will be easier to eliminate $y$.

Step 2: Make the coefficients of $y$ the same

To eliminate $y$, we need to manipulate the second equation so that the coefficient of $y$ matches the first equation. In Equation 2, the coefficient of $y$ is -1, so we will multiply it by 3 to match the coefficient of $y$ in Equation 1.

Multiplying Equation 2 by 3: $3(4x - y) = 3(6)$ $12x - 3y = 18 \quad \text{(New Equation 3)}$

Now we have:

1. $2x - 3y = 8$ (Equation 1)
2. $12x - 3y = 18$ (New Equation 3)

Step 3: Subtract the equations to eliminate $y$

Now, subtract Equation 1 from New Equation 3: $(12x - 3y) - (2x - 3y) = 18 - 8$ Simplifying: $10x = 10$

Step 4: Solve for $x$

$x = \frac{10}{10} = 1$

Step 5: Substitute $x = 1$ into one of the original equations

Now that we have $x = 1$, substitute this value into either of the original equations to solve for $y$. Let's use Equation 2: $4x - y = 6$ Substitute $x = 1$: $4(1) - y = 6$ $4 - y = 6$ Solving for $y$: $-y = 6 - 4$ $-y = 2$ $y = -2$

Final Solution:

The solution to the system of equations is: $x = 1, \quad y = -2$

Do you want more details or have any questions?

Here are 5 related questions for practice:

1. Solve the system of equations using substitution: $3x + y = 9$ and $2x - y = 4$.
2. Solve using elimination: $5x + 4y = 20$ and $3x + 2y = 12$.
3. What is the geometric interpretation of solving a system of linear equations?
4. How do you solve a system of three equations with three variables using elimination?
5. Solve for $x$ and $y$ using elimination: $7x - 5y = 2$, $4x + 5y = 18$.

Tip: When using elimination, aim to match coefficients to make one variable cancel out easily.