When you need to solve systems of linear equations, there are two popular methods: substitution and elimination. Each method works differently and might be better to use in different situations.

Substitution Method:

1. Start by solving one equation for one variable.

   • For example, take these two equations:
     • ( y = 2x + 3 )
     • ( 3x + y = 9 ) Here, we can solve for ( y ) in the first equation.

2. Next, substitute that expression into the other equation.

   • So we plug ( y ) into the second equation to get:
     • ( 3x + (2x + 3) = 9 ).

3. Now, solve for the variable ( x ), and then use that value to find the other variable ( y ).

   • From the equation ( 5x + 3 = 9 ), we find ( x = 1.2 ). Then we go back and find ( y ): ( y = 5.4 ).

Elimination Method:

1. First, write the equations so they are lined up neatly, which helps to eliminate one variable by either adding or subtracting the equations.

   • For example, consider these equations:
     • ( 2x + 3y = 6 )
     • ( 4x + 6y = 12 ). Notice that the second equation is just a double of the first one, which makes elimination trickier here.

2. If needed, you can change the equations (like multiplying) to help line up the variables for elimination.

   • If the equations were different, you might multiply one to match the coefficients.

3. Finally, add or subtract the equations to cancel out one variable and solve for the remaining one.

Each method has its strengths depending on the equations. Practicing both methods is a great way to get comfortable with solving systems of equations!