use elimination method to solve {6y=18+12x} and {2x=-y+-6}

We are given the following system of equations:

1. \( 6y = 18 + 12x \)
2. \( 2x = -y - 6 \)

### Step 1: Rearrange both equations into standard form
We want to express both equations in the form \( ax + by = c \).

#### Equation 1: 
\[
6y = 18 + 12x
\]
Subtract \(12x\) from both sides:
\[
6y - 12x = 18 \quad \text{(Equation 1)}
\]

#### Equation 2:
\[
2x = -y - 6
\]
Add \( y \) to both sides:
\[
2x + y = -6 \quad \text{(Equation 2)}
\]

Now, the system is:
1. \( 6y - 12x = 18 \)
2. \( 2x + y = -6 \)

### Step 2: Eliminate one variable
We can eliminate \( y \) by multiplying the second equation by 6 to match the coefficient of \( y \) in the first equation.

Multiply equation 2 by 6:
\[
6(2x + y) = 6(-6)
\]
This simplifies to:
\[
12x + 6y = -36 \quad \text{(Equation 3)}
\]

Now we have the system:
1. \( 6y - 12x = 18 \)
2. \( 12x + 6y = -36 \)

### Step 3: Add the equations to eliminate \( y \)
Add equation 1 and equation 3 together:
\[
(6y - 12x) + (12x + 6y) = 18 + (-36)
\]
Simplifying:
\[
6y + 6y - 12x + 12x = 18 - 36
\]
\[
12y = -18
\]
Solve for \( y \):
\[
y = \frac{-18}{12} = -\frac{3}{2}
\]

### Step 4: Substitute \( y = -\frac{3}{2} \) into one of the original equations
Substitute into equation 2:
\[
2x + y = -6
\]
Substitute \( y = -\frac{3}{2} \):
\[
2x - \frac{3}{2} = -6
\]
Add \( \frac{3}{2} \) to both sides:
\[
2x = -6 + \frac{3}{2}
\]
Convert \(-6\) to a fraction:
\[
2x = \frac{-12}{2} + \frac{3}{2} = \frac{-9}{2}
\]
Solve for \( x \):
\[
x = \frac{-9}{2} \div 2 = \frac{-9}{4}
\]

### Final Solution:
The solution to the system is:
\[
x = \frac{-9}{4}, \quad y = \frac{-3}{2}
\]

Would you like any further details or have any questions?

---

Here are 5 related questions to consider:
1. How can you solve systems of equations using substitution instead of elimination?
2. What is the advantage of using elimination over substitution in solving systems of equations?
3. How can you solve a system of three equations with three variables using elimination?
4. How do you verify that your solution is correct in a system of equations?
5. Can you apply elimination to systems of nonlinear equations?

**Tip:** Always double-check your solution by substituting the values of \(x\) and \(y\) back into the original equations.

Learn how to solve the system of linear equations 6y = 18 + 12x and 2x = -y - 6 using the elimination method. This detailed step-by-step guide covers rearranging equations, eliminating variables, and solving for x and y. Suitable for Grades 8-10.

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