A system of equations is a group of multiple equations solved simultaneously to reach the same or multiple solutions. Many systems of equations can be graphed on a coordinate plane for easy visualization and solving. However, not all systems have solutions, and some have many solutions.
Systems of equations can be used to represent the abstract language in word problems, making them easier to solve to the average math student. The defining characteristic is that all equations in the system share the same variables, which must logically represent the same value or values between equations.
Systems of equations can be solved by the substitution and combination methods. This involves adding the terms of the equations after multiplying them so that one set of terms cancels out. For example, 3x + 6y = 9 and -3x + 4y = 11 simplifies to 10y = 20, leaving y as 2. Then substituting 2 as y can be used to solve for x. In the system, x is equal to -1.
By setting the two terms to 0 and solving, it's possible to find the intercepts and draw line segments through them. Then, finding the intersection of the two lines provides the solution to the system. Should two or more lines end up parallel, there is no solution. If it is the same line, the system an infinite number of solutions. Nonlinear equations, such as parabolas and sinusoidal functions, can have two or more solutions.