Systems of Equations in Two Variables

Introduction

A system of equations consists of two or more equations made up of two or more variables such that all equations in the system are considered simultaneously. To find the unique solution to a system of equations, we must find a numerical value for each variable in the system that will satisfy all equations in the system at the same time. Some systems may not have a solution (for example, two distinct lines that are parallel to one another) and others may have an infinite number of solutions (for example, $y=0$ and $y=sin(x)$, since $sin(x)=0$ for an infinite number of x-values). In order for a linear system to have a unique solution, there must be at least as many equations as there are variables. Even so, this does not guarantee a unique solution.

Systems of equations can be solved in a number of ways. We can use elimination and/or substitution to solve for points of intersection shared by all equations in the system. We can also sometimes graph the equations to find the points shared by all equations, given that our system is graphable.

Here are some examples of systems of equations with two variables:

• One solution: $y=2x$ and $y=x+5$. By use of substitution, elimination, or graphing, we can find that the solution to this system is $(5,10)$, as this is the only point shared by both equations in the system.
• Multiple solutions: $y=x^{2}$ and $y=4$. Since $x^{2}=4$ when x = 2 or -2, we know there are two solutions to this system: (2, 4) and (-2, 4). This is also clear when we graph these two equations, as $y=4$ intersects the parabola $y=x^{2}$ when x = -2, and when x = 2.
• No solutions: $y=x^{2}$ and $y=-3$. Since $x^{2}$ is positive for all values of x, it cannot equal -3. When graphed, we can see that these two equations never intersect. Thus there are no solutions to this system of equations.
• Infinite solutions: $y=sin(x)$ and $y=0$. $sin(x)=0$ for all $x=\pi k$, where k is some integer. So, this system has an infinite number of solutions of the form $(\pi k,0)$; that is, the solution set is {...$(-2\pi ,0)$, $(-\pi ,0)$, $(0,0)$, $(\pi ,0)$, $(2\pi ,0)$...}.

Resources

• Linear Systems with Two Variables, Paul's Online Notes
• Systems of Linear Equations with Two Variables, OpenStax
• Systems of Nonlinear Equations with Two Variables, OpenStax