System of Equations

1. system of equations – A set of two or more equations with the same variables. 2. substitution – Using algebraic methods to find an exact solution of a system of equations. Key Concept: Systems of Equations  A system of equations is a collection of two or more equations with the same set of variables.  The solution of a system of equations is where the graphs of the equations intersect.  The solution of a system of equations that intersect at one point is an ordered pair that is a solution of both equations. For example, the graph at the right shows the graphs of the equations 𝑦 = 𝑥 + 1 and 𝑦 = 2𝑥 − 2. The graphs appear to intersect at the point (3, 4). To confirm that this is the solution of the system, we can replace (𝑥, 𝑦) with (3, 4) in both equations: 𝑦 = 𝑥 + 1 𝑦 = 2𝑥 − 2 Key Concept: Solving Systems by Graphing  One way to solve a system of equations is to graph the equations on the same coordinate plane.  As we can see from the example above, the coordinates of the point where the graphs intersect is the solution of the system. Examples: 1. Solve the system of equations by graphing. 𝑦 = 2𝑥 𝑦 = 𝑥 + 3  A system of equations can have one solution, no solution, or infinitely many solutions. Graph Intersecting Lines Parallel Lines Same Line Number of Solutions one solution no solution infinitely many solutions You can also use algebraic methods to solve a system of equations. One method is called substitution.  To solve systems of equations by substitution, solve one equation for a variable. Then, replace the variable in the second equation with its equivalent from the first equation.