Solving Systems of Equations

The key to solving systems of equations is following a clear process. Here's the 5-step method that works every time:

1. Line up your equations so they're easy to work with
2. Multiply equations if needed to make coefficients match
3. Add or subtract the equations (do the opposite operation of what you need)
4. Solve for one variable first
5. Plug it back in to find the other variable

Let's see this in action! In the example $7x + 2y = 10$ and $4x + 3y = 15$, we first lined up the equations. When we added them together, we got $5y = 25$, which simplified to $y = 5$. Then we plugged this value back into $4x + 3y = 15$ and found that $x = 0$. Our solution is the point $(0,5)$.

Remember this! When adding or subtracting equations, you're trying to eliminate one variable so you can solve for the other. Choose your operations carefully to make one variable disappear.

For another system like $-3x + y = 2$ and $-7x + y = -6$, you would follow the same steps, looking to eliminate the $y$ variable since its coefficients are the same.