Chapter 6 Systems of Linear/Nonlinear Equations

Up until now, when we concerned ourselves with solving different types of equations there was only one equation to solve at a time. Given an equation [latex]f(x) = g(x)[/latex], we could check our solutions geometrically by finding where the graphs of [latex]y=f(x)[/latex] and [latex]y=g(x)[/latex] intersect. The [latex]x[/latex]-coordinates of these intersection points correspond to the solutions to the equation [latex]f(x) = g(x)[/latex], and the [latex]y[/latex]-coordinates were largely ignored. If we modify the problem and ask for the intersection points of the graphs of [latex]y=f(x)[/latex] and [latex]y=g(x)[/latex], where both the solution to [latex]x[/latex] and [latex]y[/latex] are of interest, we have what is known as a system of equations, written as \[ \left\{ \begin{array}{rcl} y & = & f(x) \\ y & = & g(x) \\ \end{array} \right.\] The curly bracket notation means we are to find all pairs of points [latex](x,y)[/latex] which satisfy both equations.