2.1 Solving Systems of Equations in Two Variables. Objectives Solve systems of equations graphically Solve systems of equations algebraically Page 67

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1 2.1 Solving Systems of Equations in Two Variables Objectives Solve systems of equations graphically Solve systems of equations algebraically Page 67

2 System of Equation A system of equations is a collection of two or more equations with a same set of unknowns. In solving a system of equations, we try to find values for each of the unknowns that will satisfy every equation in the system. The ordered pair is called the solution.

3 A consistent system is a system that has at least one solution. An inconsistent system is a system that has no solution. The equations of a system are dependent if ALL the solutions of one equation are also solutions of the other two equations. In other words, they end up being the same line. The equations of a system are independent if they do not share ALL solutions. They can have one point in common, just not all of them.

4 Three possible types of solutions: Independent system: one solution and one intersection point Inconsistent system: no solution and no intersection point Dependent system: the solution is the whole line

5 Solve the following system by graphing. 2x 3y = 2 4x + y = 24

6 Solve the following system by graphing. y = 36 9x 3x + y/3 = 12

7 7x + 2y = 16 21x 6y = 24 Solve the following system by graphing.

8 Solve the following system by graphing. Solve the following system by substitution. 2x 3y = 2 4x + y = 24 Then the solution is (x, y) = (5, 4).

9 Solve the following system by y = 36 9x 3x + y/3 = 12 substitution. solution: y = 36 9x

10 Solve the following system using 2x + y = 9 3x y = 16 addition. Then the solution is (x, y) = (5, 1). x 2y = 9 x + 3y = 16 Then the solution is (x, y) = (1, 5).

11 Homework Assignment on the Internet Lesson 1: Section 2.1(Read Solving Systems of Equations in Two Variables): Pp 71-72: even, 34.

12 2.2 Solving Systems of Equations in Three Variables Objective Solve systems of equations involving three variables algebraically. Page 73

13 One Solution If the system in three variables has one solution, it is an ordered triple (x, y, z) that is a solution to ALL THREE equations. In other words, when you plug in the values of the ordered triple, it makes ALL THREE equations TRUE. If you do get one solution for your final answer, is this system consistent or inconsistent? If you said consistent, you are correct! If you do get one solution for your final answer, would the equations be dependent or independent? If you said independent, you are correct!

14

15 No Solution If the three planes are parallel to each other, they will never intersect. This means they do not have any points in common. In this situation, you would have no solution. If you get no solution for your final answer, is this system consistent or inconsistent? If you said inconsistent, you are right! If you get no solution for your final answer, would the equations be dependent or independent? If you said independent, you are correct!

16

17 Infinite Solutions If the three planes end up lying on top of each other, then there is an infinite number of solutions. In this situation, they would end up being the same plane, so any solution that would work in one equation is going to work in the other. If you get an infinite number of solutions for your final answer, is this system consistent or inconsistent? If you said consistent you are right! If you get an infinite number of solutions for your final answer, would the equations be dependent or independent? If you said dependent you are correct!

18

19 Solving Systems of Linear Equations in Three Variables Using the Elimination Method Step 1: Simplify and put all three equations in the form Ax + By + Cz = D if needed. Step 2: Choose to eliminate any one of the variables from any pair of equations. Step 3: Eliminate the SAME variable chosen in step 2 from any other pair of equations, creating a system of two equations and 2 unknowns. Step 4: Solve the remaining system found in step 2 and 3. Step 5: Solve for the third variable. Step 6: Check.

20 Solving Systems of Linear Equations in Three Variables Using the Elimination Method (3/4, -2, 1/2) is a solution to our system.

21 Solving Systems of Linear Equations in Three Variables Using the Elimination Method You have an infinite number of solutions.

22 The answer is no solution. Solving Systems of Linear Equations in Three Variables Using the Elimination Method

23 Homework Assignment on the Internet Lesson 2: Section 2.2(Read Solving Systems of Equations in Three Variables): Pp 76-77: 8 22 even

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