Understanding Systems of Linear Equations

A system of linear equations is made up of two or more simple equations. These equations show different relationships or rules. Each equation can be written like this:

[ y = mx + b ]

In this equation:

• ( m ) is the slope, which tells us how steep the line is.
• ( b ) is the y-intercept, where the line crosses the y-axis.

How Individual Equations Relate to the Whole System

1. Variables: Systems usually involve different variables. For example, we often see ( x ) and ( y ).

2. Solutions: A solution to the system is a pair of numbers, like ( (x, y) ). This pair makes all the equations true at the same time.

3. Interdependence: The solutions of the individual equations show where the lines meet on a graph.

What You Can Find in a System

A system can have:

• One Solution: This happens when the lines meet at one point.

• No Solution: This means the lines are parallel and never cross.

• Infinitely Many Solutions: In this case, the lines lay on top of each other.

By understanding these ideas, we can better work with systems of linear equations!