When you’re solving systems of linear equations, there are two popular methods you can use: substitution and elimination. Let’s break down how each method works.

Substitution Method:

1. Solve for One Variable: Start with one equation and solve for one variable. For example, if you have the equation $y = 2x + 1$, you’ve found $y$ in terms of $x$.

2. Substitute: Take that expression and put it into the other equation. If your other equation is $3x + 2y = 12$, replace $y$ with $2x + 1$.

3. Solve: Now, you’ll just have one variable to solve for. Once you find its value, plug it back into the first equation to find the other variable.

Elimination Method:

1. Align Equations: Make sure both equations are in a standard form, like $Ax + By = C$.

2. Make Coefficients Opposite: Change the equations so that adding or subtracting them will get rid of one variable. You might have to multiply one or both equations by a number to do this.

3. Add or Subtract: Now, combine the equations to solve for one variable. After that, substitute the value you found back into one of the original equations to find the other variable.

As for finding solutions, you might come across:

• Consistent: There is one unique solution (the lines intersect).
• Inconsistent: There is no solution (the lines are parallel).
• Dependent: There are infinitely many solutions (the lines overlap).

Understanding these methods can make it much easier to work with systems of equations!