Systems of Different Functions

A system of equations is a set of two or more equations that you solve at the same time. In this lesson, we look at a linear equation (a straight line like $y = mx + b$) and a quadratic equation (a parabola like $y = ax^2 + bx + c$). The solution to this system is the set of $(x, y)$ coordinates where the line and the curve actually touch or cross. Unlike two lines, which usually meet at just one point, a line and a parabola can cross at two points, touch at one point, or never meet at all (zero solutions).

When you set the equations equal and move everything to one side, you get a new quadratic equation in the form $ax^2 + bx + c = 0$. You can use the discriminant ($b^2 - 4ac$) to predict how many times the line and parabola will cross without even graphing them! If $b^2 - 4ac > 0$, there are two solutions. If it equals $0$, there is one solution. If it is less than $0$, the line and parabola never meet.

While algebra gives us exact answers, graphing calculators (like Desmos or a TI-84) are vital for complex systems. When you graph both equations, the solutions are the points of intersection. This is especially helpful when the numbers are 'messy' decimals. In a simulation, if you see a line passing through a parabola, the $x$-value usually represents time or distance, while the $y$-value represents height or cost.