System of Linear Equations in 2 variables

How to Solve Systems

A system of linear equations means two or more linear equations. (In plain speak: 'two or more lines') If these two linear equations intersect, that point of intersection is called the solution to the system of linear equations.

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:

picture of system of equations

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.

picture of system of equations

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 exactly 1 time.

picture of system of equations

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

no solution pic
Case 3: 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

infiinite solutions
Example 1

The solution of the system of equations on the left is (2, 2) which marks the point where the two lines intersect.


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:

on Solutions of Systems of Equations

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