Infinite Solutions or No Solution

Infinite Solutions

When two lines have infinite solutions, they’re the same line! This means they also share the same equation.

Example:

$$2y+4x=8\6y+hx=24$$

The system of equations above has infinite solutions. What is the value of h?

A) 4
B) 6
C) 8
D) 12

Strategy:

  1. Get both equations into (y=mx+b) form
  2. Set the right sides equal and solve for the constant.

Or

  1. Notice if one equation can be multiplied up to match the other equation.
  2. Solve for the constant.

Here is what infinite solutions looks like on a graph: they’re the same line!

No Solutions

Systems of equations with no solutions are parallel lines – they have the same slope and different y-intercepts.

If you solve for a system and end up with something like 5 = 3, assuming you did your math correctly, this incorrect equation means no solution.

Example:

$$2x=5y-8\3y+1=kx$$

The system of equations above has no solutions. What is the value of k?

A) (frac25)
B) (frac65)
C) (frac13)
D) (6)

Strategy:

  1. Get both equations in (y=mx+b).
  2. Set the slopes equal and solve for the constant.

Here’s an example of what no solutions looks like on a graph:

Traps

Constants
The SAT will often use constants with systems of equations. Just remember that they’re only numbers, and you should do the same math you would if they looked like a number.