1. Systems of Equations Section 4.1 MATH 116 - 460 Mr. Keltner

2. Systems of Equations A system of two linear equations with two variables x and y, consists of two equations that can be written in the following form: Also called a linear system. A solution of a system of linear equations is an ordered pair (x, y) that satisfies each equation. –Solutions correspond to points where the graphs of the equations intersect. –Is only a solution if it makes ALL equations true.

3. Intersection Points Intersect at one point (x, y) Never intersect--parallel lines Intersect at Infinitely many points (x, y) Any two lines we consider could fall under one of these scenarios: –The lines intersect at one specific point –The lines do not intersect (they are parallel lines) –The lines intersect everywhere they exist (the equations describe the same graph)

4. Braces!! A system of equations is connected by a brace to show that we are to find the value(s) that are solutions to both equations. There are three methods we learn about to solve a system of equations. –Substitution Method –Elimination Method –Graphically

5. Checking a Solution 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.

6. Graphical Method We can use a graphing device to solve the system of equations. –We do not need a calculator, though. –We can always sketch the graphs by making a table of values or plotting the x- and y-intercepts. On most graphing devices, it is necessary to express the equation in terms of y = f(x) before it can be graphed. Not all equations we encounter can easily be rearranged in this form, so this method may not work for every system of equations.

7. Graphical Method Steps 1.Graph each equation. Express each equation in a form suitable for the graphing calculator by solving for y as a function of x. Graph the equations on the same screen. 2.Find the intersection points. The solutions are the x- and y- coordinates of the points of intersection. 3.Check your solution(s). Example 2: Find the solution of the system. y x 5 5 -5

8. Example 3 Find the solution of the system. y x 5 5 -5

9. Example 4 Find the solution of the system. y x 5 5 -5

10. Classifying Systems of Equations Systems of equations are either consistent or inconsistent. –Inconsistent--a system of equations that has no solution. –Consistent--a system of equations that has at least one solution. Consistent systems are further classified as either dependent or independent systems. Dependent systems are equations with identical graphs. (Infinitely many solutions) Independent systems are equations that have different graphs. (One unique solution)

11. Example 5: Classifying Systems Determine whether each system graphed below is: A.Consistent with dependent equations B.Consistent with independent equations C.Inconsistent with independent equations

12. Substitution Method With the substitution method, we solve one equation for a single variable and substitute into the other equation. Use the steps on the next slide to solve the system of equations below

13. Substitution Method Steps 1.Solve for one variable. Choose an equation where a variable is already isolated or can easily be isolated. 2.Substitute. Substitute the expression from Step 1 into the other equation and solve for the only remaining variable. 3.Back-Substitute. Substitute the value you found in Step 2 back in the expression from Step 1 to solve for the remaining variable. 4.Check your answer. Example 6: Find all solutions to the system.

14. Example 7 Find all solutions of the system.

15. 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.

16. Elimination Method Steps 1.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) 2.Add the Equations. Add the two equations to eliminate one variable, then solve the resulting equations. 3.Back-Substitute. Substitute the value you found in Step 2 back into one of the original equations, and solve for the remaining variable. 4.Check your answer. Example 8: Find all solutions to the system.

17. Example 9 Find all the solutions of the system.

18. 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.

19. Which Method To Use? Features of the systemSuggested 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

20. 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: A.Substitution Method B.Elimination Method C.Graphical Method

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