Introduction to Systems of Equations

Khan Academy: Introduction to Systems of Equations

Stations: Unit 7 Stations

Identifying a Solution (Skill #1)

Which ordered pair satisfies both equations?

Infinitely Many Solutions

When two expressions are equivalent, all ordered pairs will satisfy both equations.


y=10-x       and      y+x=10

The expression on the right is just a rearrangment of the expression on the left.

Using the relation y=10-x, we can generate the following ordered pairs:

    \[\{(0,10); (1, 9); (2,8); (5,5); (10,0); (11,-1)\}\]


Notice that all of those ordered pairs also satisfy y+x=10.

As there are infinitely many points on any line, there are infinitely many ordered pairs that satisfy the system

    \[\begin{cases}y=10-x\\ y+x=10 \end{cases}\]

If one equation can be rearranged to the other, the equations are equivalent. All ordered pairs satisfying one equation will also satisfy the other.

No Solutions

If our set of equations leads to two lines that are parallel but not coincident, then there is no ordered pair that will satisfy both relations.


On a graph:

Desmos Activities: Go to, ask for the class code.

  • Polygraph (requires pairs of students)
  • Systems of Two Linear Equations (requires at least 4 students at a time)

Maths is Fun Page:

System of Equations Menu