Number of Solutions

Two solutions – the straight line intersects the parabola at two points

One solution – the straight line is tangent to the parabola

No solutions – the line and parabola don’t touch

A common question type for non-linear systems is:

$$y=x^2+3x–7$$

$$2y-10x+16=0$$

How many solutions are there to the system of equations above? 

A) There are exactly 4 solutions. 
B) There are exactly 2 solutions. 
C) There is exactly 1 solution. 
D) There are no solutions. 

Strategy:

  1. Graph the equations in Desmos and find the intersection(s).

Desmos, shows us that the two equations intersect at the point ((1,-3)). There are no other points of intersection for the two graphs. This tells us that there is ONLY ONE solution to the system of equations, so choice C is the correct answer.

One last example:

A system of three equations and their graphs in the xy-plane are shown above. How many solutions does the system have? 

A) One
B) Two
C) Three
D) Four

They are trying to confuse us! This looks very complicated, but again, all we do is look at where all three lines intersect.