use elimination method to solve {5x+2y=-19} and {3x-4y=-27}

We are given the system of equations:

\[
1) \, 5x + 2y = -19
\]
\[
2) \, 3x - 4y = -27
\]

To solve this system using the **elimination method**, we want to eliminate one of the variables by making their coefficients in both equations equal, then subtract or add the equations to eliminate that variable.

### Step 1: Make the coefficients of \(y\) equal
To eliminate \(y\), we can multiply the first equation by 2, and the second equation by 1, so that both \(y\)-terms will have the same absolute value but opposite signs.

\[
(1) \times 2 \Rightarrow 10x + 4y = -38
\]
\[
(2) \times 1 \Rightarrow 3x - 4y = -27
\]

Now, we have:
\[
10x + 4y = -38
\]
\[
3x - 4y = -27
\]

### Step 2: Add the two equations
Adding these two equations will eliminate \(y\):

\[
(10x + 4y) + (3x - 4y) = -38 + (-27)
\]

This simplifies to:

\[
13x = -65
\]

### Step 3: Solve for \(x\)
Now, solve for \(x\):

\[
x = \frac{-65}{13} = -5
\]

### Step 4: Substitute \(x = -5\) into one of the original equations
We can substitute \(x = -5\) into the first equation \(5x + 2y = -19\) to find \(y\):

\[
5(-5) + 2y = -19
\]
\[
-25 + 2y = -19
\]
\[
2y = -19 + 25
\]
\[
2y = 6
\]
\[
y = \frac{6}{2} = 3
\]

### Final Solution:
The solution to the system of equations is:
\[
x = -5, \, y = 3
\]

Would you like more details on any of these steps, or do you have any questions?

Here are 5 related questions you might explore next:
1. How would the elimination method change if we had different coefficients?
2. Can you use the substitution method to solve the same system?
3. What happens if the system has no solution or infinitely many solutions?
4. How can you graphically interpret the solution to this system?
5. What is the determinant of the coefficient matrix for this system?

**Tip:** The elimination method is especially useful when both variables have small integer coefficients.

Learn how to solve the system of linear equations 5x+2y=-19 and 3x-4y=-27 using the elimination method. This detailed step-by-step guide explains how to eliminate variables and solve for x and y, with a solution for x = -5 and y = 3. Suitable for Grades 9-10.

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