1. Systems of Equations Graphing Systems of Equations is used by people with careers in biological science, such as ecologists. An ecologist uses graphs of systems to study populations of organisms and how they relate to their environment.

2. Vocabulary The equations x + y = 24 and 3x + y = 44 together are called a System of Equations. The equations x + y = 24 and 3x + y = 44 together are called a System of Equations. One method for solving a system of equations is to carefully graph the equations on the same coordinate plane. The coordinates of the point at which the graphs intersect is the solution of the system. One method for solving a system of equations is to carefully graph the equations on the same coordinate plane. The coordinates of the point at which the graphs intersect is the solution of the system.

3. A system of two linear equations has exactly one ordered pair as its solution when the graphs of the equations intersect at exactly one point. (-2,2)

4. What if… the two linear equations are parallel ? (there will be no solutions) the two linear equations are the same line? (there will be infinitely many points)

5. Another method is called Substitution Let’s do a problem. Let’s do a problem. If 3 bags of apples and 2 bags of oranges cost $3.10 and 2 bags of apples and 4 bags of oranges cost $4.20, how much does each bag of apples and each bag of oranges cost? If 3 bags of apples and 2 bags of oranges cost $3.10 and 2 bags of apples and 4 bags of oranges cost $4.20, how much does each bag of apples and each bag of oranges cost? Bag of apples = x Bag of apples = x Bag of oranges = y Bag of oranges = y

6. If 3 bags of apples and 2 bags of oranges cost $3.10 and 2 bags of apples and 4 bags of oranges cost $4.20, how much does each bag of apples and each bag of oranges cost? If 3 bags of apples and 2 bags of oranges cost $3.10 and 2 bags of apples and 4 bags of oranges cost $4.20, how much does each bag of apples and each bag of oranges cost? 3x + 2y = 3.12x + 4y = 4.2 3x + 2y = 3.12x + 4y = 4.2 Solve one equation for one variable. Solve one equation for one variable. 3x + 2y = 3.1 3x + 2y = 3.1 -3x -3x -3x -3x 2y = -3x + 3.1 2y = -3x + 3.1 2 2 2 2 2 2 y = -1.5x + 1.55 y = -1.5x + 1.55

7. If 3 bags of apples and 2 bags of oranges cost $3.10 and 2 bags of apples and 4 bags of oranges cost $4.20, how much does each bag of apples and each bag of oranges cost? y = -1.5x + 1.55 3x + 2y = 3.1 2x + 4y = 4.2 3x + 2y = 3.1 2x + 4y = 4.2 2x + 4( y ) = 4.2 2x + 4( y ) = 4.2 2x + 4(-1.5x + 1.55) = 4.2 2x + 4(-1.5x + 1.55) = 4.2 2x + -6x + 6.20 = 4.2 2x + -6x + 6.20 = 4.2 -4x + 6.20 = 4.2 -4x + 6.20 = 4.2 - 6.20 -6.20 - 6.20 -6.20 -4x = -2 -4x = -2 -4 -4 -4 -4 x =.5 x =.5

8. If 3 bags of apples and 2 bags of oranges cost $3.10 and 2 bags of apples and 4 bags of oranges cost $4.20, how much does each bag of apples and each bag of oranges cost? X =.5 3x + 2y = 3.1 2x + 4y = 4.2 3x + 2y = 3.1 2x + 4y = 4.2 3(.5) + 2y = 3.1 3(.5) + 2y = 3.1 1.5 + 2y = 3.1 1.5 + 2y = 3.1 -1.5 -1.5 -1.5 -1.5 2y = 1.6 2y = 1.6 2 2 2 2 y =.8 y =.8

9. If 3 bags of apples and 2 bags of oranges cost $3.10 and 2 bags of apples and 4 bags of oranges cost $4.20, how much does each bag of apples and each bag of oranges cost? So we found out that So we found out that x =.5 or $.50 for a bag of apples x =.5 or $.50 for a bag of apples and y =.8 or $.80 for a bag of oranges. and y =.8 or $.80 for a bag of oranges.

10. Here’s the graph! (.5 (.5,.8 ) 3x+2y=3.40 y=-1.5x + 1.55 2x+ 4y = 4.20 y=-.5x + 1.05

11. Quick Review When two linear equations intersect you have an ordered pair. When two linear equations intersect you have an ordered pair. This ordered pair is the solution for the Systems of Equations. This ordered pair is the solution for the Systems of Equations. You can graph the two equations and see where they intersect. You can graph the two equations and see where they intersect. You can also use Substitution and find the solution without graphing. You can also use Substitution and find the solution without graphing. It’s always a good idea to check solutions to make sure we didn’t make a mistake. It’s always a good idea to check solutions to make sure we didn’t make a mistake.

12. I hear there is another way to find the solution to a system of equations. I hear there is another way to find the solution to a system of equations. It’s called Elimination It’s called Elimination But that’s another Power Point. But that’s another Power Point.