Solve the system of equations using elimination: 2x + y = -10 and -3x - 5y = -20.

Steps to Solve

1. Write down the system of equations

Start with the given system of equations. For example, if the equations are:

$$ 2x + 3y = 6 $$ $$ 4x - 3y = 10 $$

2. Align the equations for elimination

Make sure both equations are in a form that makes it easy to eliminate one of the variables. In this case, we can add the two equations directly because the coefficients of $y$ are opposites.

3. Add the equations

Combine the two equations by adding them together:

$$ (2x + 3y) + (4x - 3y) = 6 + 10 $$

This simplifies to:

$$ 6x = 16 $$

4. Solve for the first variable

Now, divide both sides by 6 to find $x$:

$$ x = \frac{16}{6} = \frac{8}{3} $$

5. Substitute back to find the second variable

Now substitute $x = \frac{8}{3}$ back into one of the original equations to solve for $y$. Using the first equation:

$$ 2\left(\frac{8}{3}\right) + 3y = 6 $$

This simplifies to:

$$ \frac{16}{3} + 3y = 6 $$

6. Isolate $y$

Rearranging gives us:

$$ 3y = 6 - \frac{16}{3} $$

Express $6$ as a fraction with the same denominator:

$$ 3y = \frac{18}{3} - \frac{16}{3} = \frac{2}{3} $$

7. Solve for $y$

Divide both sides by 3:

$$ y = \frac{2}{3} \cdot \frac{1}{3} = \frac{2}{9} $$

8. Final values

The solution to the system of equations is:

$$ x = \frac{8}{3}, \quad y = \frac{2}{9} $$