To solve systems of equations with linear functions, you can use one of three methods: graphing, substitution, or elimination. Let’s break them down into simpler steps!

1. Graphing

First, you will graph each equation on the same set of axes. The point where the lines cross is the solution to the system.

For example, take these two equations:

1. (y = 2x + 1)
2. (y = -x + 4)

When you graph both lines, you will find the point where they intersect. That point gives you the values of (x) and (y) that solve the system.

2. Substitution

In this method, you'll solve one equation for one variable and then put that value into the other equation. Let’s use the same equations as before:

1. From (y = 2x + 1), you know what (y) is.

2. Substitute (y) into the second equation:

   (2x + 1 = -x + 4)

Now, solve for (x). Once you find (x), put that back into one of the original equations to find (y).

3. Elimination

This method helps you get rid of one variable by adding or subtracting the equations. Here’s how you do it:

1. If needed, multiply the second equation to make the numbers in front of one variable match.
2. Add or subtract the equations to eliminate one variable. Then solve for the variable that’s left.

After you find both variables using any of the methods, you will have the solution for the system. Remember to check your solution by plugging the values back into the original equations!