solve the system with the addition method:
-4x-5y=-20
-6x+5y=70

To solve the system of equations using the **addition method**, also known as the elimination method, we aim to eliminate one of the variables by adding the two equations together. Let's go through the steps:

The system of equations is:

\[
1) \ -4x - 5y = -20
\]
\[
2) \ -6x + 5y = 70
\]

### Step 1: Add the two equations

Notice that the terms involving \(y\) have opposite signs (\(-5y\) and \(+5y\)), so adding the two equations will eliminate the \(y\)-terms:

\[
(-4x - 5y) + (-6x + 5y) = (-20) + 70
\]

Simplifying both sides:

\[
-4x - 6x = -20 + 70
\]

\[
-10x = 50
\]

### Step 2: Solve for \(x\)

Divide both sides by \(-10\):

\[
x = \frac{50}{-10} = -5
\]

### Step 3: Substitute \(x = -5\) into one of the original equations

We can substitute \(x = -5\) into either of the original equations. Let's use the first equation:

\[
-4x - 5y = -20
\]

Substitute \(x = -5\):

\[
-4(-5) - 5y = -20
\]

Simplify:

\[
20 - 5y = -20
\]

### Step 4: Solve for \(y\)

Subtract 20 from both sides:

\[
-5y = -20 - 20
\]

\[
-5y = -40
\]

Now divide both sides by \(-5\):

\[
y = \frac{-40}{-5} = 8
\]

### Solution:
The solution to the system is:

\[
x = -5, \quad y = 8
\]

If you'd like any further clarifications, feel free to ask!

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Here are 5 questions related to this topic:
1. Can you solve a similar system of equations with different coefficients?
2. What happens if both \(x\) and \(y\)-terms do not cancel immediately after addition?
3. How can you solve a system if you want to eliminate \(x\) instead of \(y\)?
4. What if one equation needs to be multiplied by a constant first in the elimination method?
5. Can you verify the solution by plugging \(x = -5\) and \(y = 8\) back into both equations?

**Tip:** Always check your solution by substituting the values of \(x\) and \(y\) back into both original equations to ensure accuracy.

solve the system with the addition method: -4x - 5y = -20 and -6x + 5y = 70

Learn how to solve the system of linear equations -4x - 5y = -20 and -6x + 5y = 70 using the addition method. This step-by-step guide covers key algebraic techniques, including the elimination of variables and substitution. Suitable for students in Grades 8-10.

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