Lesson Summary A system of linear equations can have a unique solution, no solution, or infinitely many solutions. Systems with a unique solution are comprised of two linear equations whose graphs have different_ that is, their graphs in a coordinate plane will be two distinct lines that intersect at only one point. Systems with no solutions are comprised of two linear equations whose graphs have the same points; that is, their graphs in a coordinate plane will be two parallel lines (with no intersection). Systems with infinitely many solutions are comprised of two linear equations whos raphs have the same and the same raphs in a coordinate plane are the same line (i.e., every solution to one equation v a solution to the other equation).

Did you know that the study of systems of linear equations dates back to ancient civilizations? The Babylonians were among the first to solve simple linear equations as far back as 2000 BC! They used geometric methods and numerical solutions, laying the groundwork for modern algebra. Fast forward to the Renaissance, when mathematicians like René Descartes helped formalize these concepts, making it easier for us to visualize and solve systems of equations today. In real-world scenarios, the application of systems of linear equations is abundant! For example, in business, companies often need to solve problems involving supply and demand. Imagine a bakery planning to maximize profit by determining how many cupcakes and cookies to bake, taking into account limited resources like flour and sugar—solving a system of equations can help them find that perfect balance!