Systems of Equations

Systems of Equations A system of linear equations consists of two or more equations A solution is an ordered pair (x, y) that satisfies each equation. Solution is the point of intersection. Only a solution if it makes ALL equations true.

Intersection Points Three scenarios: The lines intersect at one specific point The lines do not intersect (parallel lines) The lines intersect everywhere they exist (same graph)

Classifying Systems of Equations Systems of equations are either consistent or inconsistent. Inconsistent--a system of equations that has no solution (parallel lines). Consistent--a system of equations that has at least one solution. Consistent Dependent systems =Infinitely many solutions Consistent Independent systems = One unique solution

Methods There are four methods: Graphing Substitution Method Elimination Method Matrices

Checking a Solution Example 1: For the system of equations We can check a particular point in the coordinate plane to see if it is a solution to the system of equations. For an ordered pair to be a solution to the system of equations, it must be a solution to both equations of the system. Example 1: For the system of equations determine whether the point (4, -1) is a solution to the system. Answer: (4, -1) is a solution to the system of equations.

Graphical Method Steps Example 2: Find the solution of the system. Graph each equation. In slope-intercept form Find the intersection points. (x, y) Check your solution(s). y x 5 -5 Answer: (-1, -2)

Example 3 Find the solution of the system. Answer: No Solutions y x -5

Example 4 Find the solution of the system. 5 -5 Answer: Infinitely Many Solutions

Substitution Method Solve one equation for a variable and SUBSTITUTE into the other equation.

Example 7 Find all solutions of the system. Answer: (3, 2)

Solve by Graph: 1) 4) 2) 5) 3) 6)

Elimination Method In solving a system of equations, we are looking for the point(s) the equations have in common. By this reasoning, we are allowed to treat x and y as if they are like terms in each of the two equations. The elimination method takes advantage of this feature and works to narrow down the system by eliminating one of the variables.

Elimination Method Steps Adjust the Coefficients. Multiply one or more of the equations by appropriate numbers so that the coefficient of one variable in one equation is the opposite of the coefficient in the other equation. (2x and -2x, for example) Add the Equations. Add the two equations to eliminate one variable, then solve the resulting equations. Back-Substitute. Substitute the value you found in Step 2 back into one of the original equations, and solve for the remaining variable. Check your answer. Example 8: Find all solutions to the system. Answer: (4, 1)

Example 9 Find all the solutions of the system. Answer: (3, -4/3)

Dependent and Independent Systems Just like we have seen with solving a single equation, we determine if there are “No Solutions” or “Infinitely Many Solutions” for systems of equations as well. Reducing to a FALSE conclusion, like -8 = 13, means that there are “No Solutions” and that the lines do not intersect. Reducing to a TRUE conclusion, like 4=4, means that there are “Infinitely Many Solutions” and that the lines coincide, or exist in the same place.

Which Method To Use? Features of the system Suggested Method One variable is isolated or can easily be isolated. Substitution Method The two equations share the same coefficient on the same variable or can be multiplied by a constant so they match. Elimination Method Each equation can easily be written in the form y = f(x). Graphical Method

Which Method To Use? For each of the systems of equations shown, choose which method would be the best option to use in solving the system: Substitution Method Elimination Method Graphical Method Answers: 1) C. Graphical, 2) A. Substitution, 3) B. Elimination

Pgs. 238-242: #’s 10-75, Multiples of 5 Assessment Pgs. 238-242: #’s 10-75, Multiples of 5