1. Systems of Equations

2. OBJECTIVES To understand what a system of equations is.
Be able to solve a system of equations from graphing the equations
Determine whether the system has one solution, no solution, or an infinite amount of solutions.
Be able to graph equations without using a graphing calculator.

3. Defining a System of Equations
A grouping of 2 or more equations, containing one or more variables.
x + y = 2
2x + y = 5
2y = x + 2
y = 5x - 7
6x - y = 5

4. How do we “solve” a system of equations???
By finding the point where two or more equations, intersect.
x + y = 6
y = 2x 6 4
Point of intersection 2 1 2

5. How do we “solve” a system of equations???
By finding the point where two or more equations, intersect.
x + y = 6
y = 2x 6 4
(2,4) 2 1 2

6. ax + by = c 2x + 3y = 6 ax + by = c -2x -2x 3y = 6 - 2x 3 y = 2 - 2 3
(Standard Form)
-2x
-2x
WE WANT THIS FORM!!!
3y = 6 - 2x 3
y = 2 - 2 3 x
y = 2 3 x
y = mx + b
(Slope- Intercept)

7. Non-Unique Solutions No Solution: when lines of a graph are parallel
since they do not intersect, there is no solution
the slopes are the same and the equation must be in slope-intercept form
Slides 10 through 14 show how I explained the different solutions to their worksheet that the students worked on in class. I didn’t focus to much on the solution they found on the worksheet but rather on the type of solution or the concepts they the solutions involved. In each slide I explain each type of solution as well as how a system has these types of solutions. Not focusing to much on the powerpoint I referred back to their worksheet to one of the examples and asked students to remember how we knew the equations graphed were parallel by only looking at the equations. From their prior knowledge students knew that it was because the equations in the system contained the same slope; making connections with new and old material.

8. Example of No Solution 2x+y=8 y=-2x-3
Remember when graphing the equations need to be in slope intercept form!!!

9. Non-Unique Solutions Infinite Solutions:
a pair of equations that have the same slope and y-intercept.
also call a Dependent System
Again giving explanations as to how we have such a solution. Just like the last slide, I also questioned students as to how we can find the whether a system has infinite solutions by looking at the equation, triggering prior knowledge.

10. Example of Infinite Solutions
2x+4y=8
4x+8y=16

11. Unique Solutions One Solution: the lines of two equations intersect
also called an Independent System
Though not a Non-Unique Solution, I explained to students that we do not call this solution as being non-unique. I included it in this slide as I wanted to students to understand the different between three types of solutions.

12. Graphing Systems in Slope-Intercept Form
2y + x = 8
y = 2x + 4
Steps:
1. Get both equations in slope-intercept form
2. Find the slope and y intercept of eq. 1 then graph
3. Find the slope and y intercept of eq. 2 then graph
4. Find point of intersection

13. Example 1: 2y+x=8 and y=2x+4
Equation 1
Equation 2
2y + x = 8
y = 2x + 4
-x x
(0,4)
m= 2/1
B=4
2y= x
The solution to the system is (0,4)
Notice the slopes and y intercepts are different so there will only be one solution
m= -1/2
b=4

14. Example 2: y=-6x+8 and y+6x=8
Equation 1
Equation 2
y=-6x+8
y+6x=8
b=8
-6x -6x
m=-6(down)
1(right)
y=8-6x
b=8
Notice both equations have the same intercept and slope. This means all the points are solutions.
m=-6(down)
1(right)
Infinite Solution

15. Example 3: x-5y=10 and -5y=-x+40
Equation 1
Equation 2
x-5y=10
-5y=-x+40
__ __ ___
-x x
-5y=10-1x
__ __ __
b=-8
m=1/5
b=-2
The slopes are the same with different intercepts so the lines are parallel
m=1/5
No Solution

16. You Try Examples
Determine whether the following equations have one, none, or infinite solutions by graphing on the graph paper provided. 1) 2) 2 3)
x + 2y = 6
y = x - 1 3
x + 2y = 8
y = 3
ANS: One Solution
(6,3)
ANS: No Solution
ANS: Infinite Solutions

17. Equation 1
Slope Intercept Form
Equation 2
8x+3y=24
Infinite Solution

18. Equation1
Equation 2
Notice the slopes are the same and the y-intercepts are different. This means that the lines are parallel so they will never intersect
No Solution

19. Equation 1
Equation 2
y=3
Notice the equations have different slopes and y-intercepts so the lines will have 1 solution
Solution (6,3)

20. Algebraically determine if a point is a solution
We must always verify a proposed solution algebraically. We propose (1,6) as a solution, so now we plug it in to both equations to see if it works:
y = 6x and y = 2x + 4,
6 = 6(1) and =2(1)+4,
(6)= 6 and = 6.
Yes, (1,6) Satisfies both equations!

21. Algebraically determine if a point is a solution
Determine if (2,12) is a solution to the system.
y=6x and y=2x+4
12=6(2) and =2(2)+4
12= and =8
Yes No
(2,12) is NOT a solution to the system