To solve the system of equations using the elimination method, we aim to eliminate one of the variables by adding or subtracting the equations.

The system of equations is:

\[
\begin{aligned}
1.\quad & 20x - 6y = 44 \\
2.\quad & -7x - 3y = 5
\end{aligned}
\]

### Step 1: Align the equations
The equations are already aligned by variables.

### Step 2: Make the coefficients of one variable equal
To eliminate \( y \), we need to make the coefficients of \( y \) equal in magnitude. The coefficients of \( y \) are \(-6\) and \(-3\), so we can multiply the second equation by 2 to make the coefficients of \( y \) match:

\[
\begin{aligned}
1.\quad & 20x - 6y = 44 \quad & \text{(original)} \\
2.\quad & 2(-7x - 3y) = 2(5) \quad & \text{(multiply by 2)}
\end{aligned}
\]

This gives us:

\[
\begin{aligned}
1.\quad & 20x - 6y = 44 \\
2.\quad & -14x - 6y = 10
\end{aligned}
\]

### Step 3: Subtract the equations
Now, subtract the second equation from the first to eliminate \( y \):

\[
(20x - 6y) - (-14x - 6y) = 44 - 10
\]

Simplifying this:

\[
20x - 6y + 14x + 6y = 34
\]

\[
34x = 34
\]

### Step 4: Solve for \( x \)
Divide both sides by 34:

\[
x = 1
\]

### Step 5: Substitute \( x \) back to find \( y \)
Substitute \( x = 1 \) into one of the original equations, say the first one:

\[
20(1) - 6y = 44
\]

Simplifying:

\[
20 - 6y = 44
\]

Subtract 20 from both sides:

\[
-6y = 24
\]

Divide both sides by \(-6\):

\[
y = -4
\]

### Solution
The solution to the system of equations is:

\[
x = 1, \quad y = -4
\]

Would you like more details or have any other questions?

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**Related questions:**

1. How can the substitution method be used to solve this system?
2. What if the coefficients were different? How would elimination change?
3. What happens if both variables have the same coefficient in elimination?
4. Can you explain how to verify the solution for both equations?
5. How does elimination compare with the graphical method of solving systems?

**Tip:** Always double-check your solution by substituting the values back into the original equations to ensure they satisfy both equations.

Learn how to solve the system of linear equations 20x - 6y = 44 and -7x - 3y = 5 using the elimination method. This problem involves algebraic concepts such as the elimination method and the linear combination theorem. Suitable for students in Grades 9-11.

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