When it comes to solving systems of linear equations, there are two popular methods: substitution and elimination. Each method has its benefits. Sometimes one method is easier to use depending on the equations you have.

Substitution Method

1. What It Is: This method involves solving one equation for one variable. Then, you take that solution and put it into the other equation.

2. Example: Imagine you have these two equations:

   $$x + y = 10$$ $$2x - y = 3$$

   First, from the first equation, we can solve for (y): [ y = 10 - x ]

   Next, we put this expression for (y) into the second equation: [ 2x - (10 - x) = 3 ]

   This simplifies to: [ 3x - 10 = 3 ]

   Solving this gives us (x = 4).

   Now, we can find (y) by plugging (x) back in: [ y = 10 - 4 = 6 ]

Elimination Method

1. What It Is: This method means you will add or subtract the equations to get rid of one variable. This makes it easier to solve for the other variable.

2. Example: Let’s use the same equations:

   $$x + y = 10$$ $$2x - y = 3$$

   Start by keeping the first equation the same:

   $$x + y = 10$$

   Now, we add these two equations to get rid of (y):

   $$3x = 13$$

   Now, solving for (x) gives us (x = \frac{13}{3}).

   Then, we can use this value to find (y) by substituting (x) back into one of the original equations.

Conclusion

• The substitution method is usually a good choice when one equation is easy to work with.
• The elimination method works well when the equations fit together nicely, making it easy to combine them.

Remember, always pick the method that seems easiest for you when solving the problem!