solve this by elimination 1: 3x +4y = -3 and 2x + 6y = 8

We can solve the system of equations by elimination.

The system of equations is:

1. \( 3x + 4y = -3 \)
2. \( 2x + 6y = 8 \)

### Step 1: Multiply the equations
We want to eliminate one variable, so we need to manipulate the equations. First, multiply both equations by factors that make the coefficients of \( x \) or \( y \) the same.

To eliminate \( x \), we can multiply:
- The first equation by 2: \( 2(3x + 4y) = 2(-3) \)
- The second equation by 3: \( 3(2x + 6y) = 3(8) \)

This gives:

1. \( 6x + 8y = -6 \)
2. \( 6x + 18y = 24 \)

### Step 2: Subtract the equations
Now, subtract equation 1 from equation 2 to eliminate \( x \):

\[
(6x + 18y) - (6x + 8y) = 24 - (-6)
\]

This simplifies to:

\[
10y = 30
\]

### Step 3: Solve for \( y \)
\[
y = \frac{30}{10} = 3
\]

### Step 4: Substitute \( y \) back into one of the original equations
Now, substitute \( y = 3 \) into one of the original equations (let's use the first one):

\[
3x + 4(3) = -3
\]
\[
3x + 12 = -3
\]
\[
3x = -3 - 12
\]
\[
3x = -15
\]
\[
x = \frac{-15}{3} = -5
\]

### Final Solution:
The solution is:
\[
x = -5, \quad y = 3
\]

Would you like further details on this process or have any questions?

Here are 5 related questions to expand on this:
1. Can you solve a similar system of equations with different coefficients using elimination?
2. How would you approach this problem if the coefficients of \( y \) were the same initially?
3. What happens if both equations are multiples of each other? How would you solve such a system?
4. Can you use substitution instead of elimination for this system? What would be the steps?
5. How can you verify that the solution is correct?

**Tip**: Always double-check your final answer by plugging the values of \( x \) and \( y \) back into the original equations to verify their correctness.

Learn how to solve the system of linear equations 3x + 4y = -3 and 2x + 6y = 8 using the elimination method. This step-by-step solution covers key algebraic concepts, elimination of variables, and solving for both x and y. Ideal for students in Grades 8-10.

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