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# System of Linear Equations in 2 variables

How to Solve Systems

#### What is a system of equations?

A system of equation just means 'more than 1 equation.'. A system of linear equations is just more than 1 line, see the picture:

#### Ok, so what is the solution of a system of equations?

The solution is where the equations 'meet' or intersect. The red point is the solution of the system.

#### How many solutions can systems of linear equations have?

There can be zero solutions, 1 solution or infinite solutions--each case is explained in detail below. Note: Although systems of linear equations can have 3 or more equations,we are going to refer to the most common case--a stem with exactly 2 lines.

Case I: 1 Solution

This is the most common situation and it involves lines that intersect at exactly one point.

Case 2: No Solutions

This only happens when the lines are parallel. As you can see, parallel lines are not going to ever meet.

Example of a stem that has no solution:

• Line 1: $$y = 5x + 13$$
• Line 2: $$y = 5x + 12$$

Case III: Infinite Solutions

This is the rarest case and only occurs when you have the same line
Consider, for instance, the two lines below (y = 2x + 1 and 2y = 4x + 2). These two equations are really the same line.

Example of a system that has infinite solutions:

• Line 1: y = 2x + 1
• Line 2: 2y = 4x + 2

#### How can we find solutions to systems of equations?

To find the solution to systems of linear equations, you can any of the methods below: