Final answer:
To solve the system of equations using linear combination, we first eliminate one variable by appropriately scaling each equation, then solve for the remaining variable. Substituting this value back into the original equations gives us the other variable. The solution to the system is x = -7 and y = -2.
Step-by-step explanation:
To solve the system of equations using linear combination (also known as the addition method), we need to find a way to eliminate one of the variables by combining the equations. Here is the given system of equations:
- -3x + 6y = 9
- 5x + 7y = -49
We can start by multiplying the first equation by 5 and the second equation by 3, with the goal of eliminating the variable x:
- 5(-3x + 6y) = 5(9) → -15x + 30y = 45
- 3(5x + 7y) = 3(-49) → 15x + 21y = -147
Now, we add the two new equations:
-15x + 30y + 15x + 21y = 45 - 147
The x terms cancel out, and we are left with:
51y = -102
Dividing both sides by 51, we get:
y = -2
Now that we have the value of y, we can substitute it into one of the original equations to find x. Let's use the first equation:
-3x + 6(-2) = 9
-3x - 12 = 9
Adding 12 to both sides gives us:
-3x = 21
Dividing by -3:
x = -7
Therefore, the solution to the system of equations is x = -7 and y = -2.