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.

Using elimination, solve 2x - 3y = 8, 4x - y = 6.

Learn how to solve the system of linear equations 2x - 3y = 8 and 4x - y = 6 using the elimination method. This step-by-step solution explains key algebraic concepts, including elimination and substitution, suitable for students in Grades 8-10.

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