6x − 12y = 24
−x − 6y = 4
find the y and x

We have the following system of equations:

1) \( 6x - 12y = 24 \)
2) \( -x - 6y = 4 \)

### Step 1: Simplify the equations if possible
Let's rewrite the first equation by dividing through by 6 to simplify:

\[
x - 2y = 4 \quad \text{(Equation 1 simplified)}
\]

The second equation can be written as it is:

\[
-x - 6y = 4 \quad \text{(Equation 2)}
\]

### Step 2: Use substitution or elimination
We will use the substitution method. From the simplified **Equation 1**:

\[
x = 2y + 4
\]

### Step 3: Substitute into the second equation
Substitute \( x = 2y + 4 \) into **Equation 2**:

\[
-(2y + 4) - 6y = 4
\]

Simplify:

\[
-2y - 4 - 6y = 4
\]

Combine like terms:

\[
-8y - 4 = 4
\]

### Step 4: Solve for \( y \)
Add 4 to both sides:

\[
-8y = 8
\]

Now divide by \(-8\):

\[
y = -1
\]

### Step 5: Substitute \( y = -1 \) into the equation for \( x \)
Substitute \( y = -1 \) into \( x = 2y + 4 \):

\[
x = 2(-1) + 4 = -2 + 4 = 2
\]

### Final Answer:
The solution is \( x = 2 \) and \( y = -1 \).

Do you want more details on any step or have further questions?

Here are 5 related questions:
1. How can we verify the solution by substituting the values back into the original equations?
2. Can we use the elimination method to solve this system instead?
3. What is the graphical interpretation of this system of equations?
4. How does the substitution method differ from other methods like matrix solutions?
5. How can systems of equations be applied in real-world scenarios?

**Tip**: Always verify your solutions by plugging the values of \(x\) and \(y\) back into both original equations.

This step-by-step guide shows how to solve a system of linear equations (6x − 12y = 24 and −x − 6y = 4) using the substitution method. It explains key algebraic concepts for students in Grades 8-10.

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