Solve the system of equations using elimination: -2x - 9y = -9 and -x - y = -8.

Steps to Solve

1. Write down the equations We start with the two equations given:

[ -2x - 9y = -9 \tag{1} ]

[ -x - y = -8 \tag{2} ]

2. Multiply the second equation to match a coefficient To eliminate (x), we can multiply equation (2) by 2:

[ 2(-x - y) = 2(-8) ]

This gives us:

[ -2x - 2y = -16 \tag{3} ]

3. Add the equations Now, we can add equation (1) and equation (3):

[ (-2x - 9y) + (-2x - 2y) = -9 + (-16) ]

This simplifies to:

[ -4x - 11y = -25 ]

4. Solve for (y) Now we can isolate (y). From equation (3), we have:

[ -2x - 2y = -16 ]

Solve for (y):

[ -2y = 2x - 16 ]

[ 2y = -2x + 16 ]

[ y = -x + 8 \tag{4} ]

5. Substitute (y) back into one of the original equations Substitute equation (4) into either of the original equations. We'll use equation (2):

[ -x - (-x + 8) = -8 ]

Simplifying this gives:

[ -x + x - 8 = -8 ]

This reduces to:

[ 0 = 0 , \text{(which holds for all x)} ]

Since it is an identity, we will substitute back into equation (1) to find the value for (x).

6. Finding (x) using one of the derived equations Using equation (4):

[ y = -x + 8 ]

Let’s pick a value for (x). If we take (x = 0):

[ y = 8 ]

So, the solution is:

[ (x, y) = (0, 8) ]