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. ax + by = c
NOTE: The equation ax + by = c is just another form of a linear equation.

8. Non-Unique Solutions No Solution: when lines of a graph are parallel
since they do not intersect, there is no solution
also called an Inconsistent System
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.

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. Non-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.

11. Examples… Determine whether the following have one, none, or infinite
solutions by looking at the slope and y-intercepts 1) 2) 3)
2y + x = 8
y = 2x + 4
y = -6x + 8
y + 6x = 8
x - 5y = 10
-5y = -x +6
ANS:
One Solution
ANS:
Infinite Solutions
ANS:
No Solution

12. Graphing Manually
Using the y-intercept and the linear slope to graph the equation:
y = 2x + 4

13. Graphing Manually
Using the y-intercept and the linear slope to graph the equation:
y = 2x + 4
1. Plot the y-intercept

14. Graphing Manually
Using the y-intercept and the linear slope to graph the equation:
y = 2x + 4
1. Plot the y-intercept
2. Use the slope to plot second point (rise and run)

15. Graphing Manually
Using the y-intercept and the linear slope to graph the equation:
y = 2x + 4
1. Plot the y-intercept
2. Use the slope to plot second point (rise and run)
If entered in the slide show, one can see that the slides provided an visual aide as to how to graph a line in y-intercept form. The powerpoint provides a perfect method for students to exactly see how they would be able to graph a line in y-interecept form giving a visual aid for each of the steps that I provide them.
3. Draw a line connecting the two points.

16. Graphing Manually cont.
Plot two points by Solving for y by entering random values for x.
2x + 3y = 4

17. Graphing Manually cont.
Plot two points by Solving for y by entering random values for x.
2x + 3y = 4
1. Enter a random value for x, then solve for y
x = 0
2(0) + 3y = 4
Just like the previous slide using equations in slope-intercept form, I give students a visual aid provided with each step that give them to graph an equation in standard form. Following each step I then display the next step on the x and y-axis next the work shown above. It gave me a good way to relate the graph back to the work I did finding the first point and vice/versa. Continue on with the slides in “Slide Show” to see the rest of the lesson.
3y = 4 3
y = 1.33
2. Plot the point: (0, 1.33)

18. Graphing Manually cont.
Plot two points by Solving for y by entering random values for x.
2x + 3y = 4
3. Enter another value for x, then solve for y
x = 2
2(2) + 3y = 4
4 + 3y = 4 -4
3y = 0
y = 0
4. Plot the point: (2, 0)

19. Graphing Manually cont.
Plot two points by Solving for y by entering random values for x.
5. Draw line connecting both points.

20. more Examples
Determine whether the following equations have one, none, or infinite solutions. If “one solution” graph it and give the point of intersection. NO GRAPHER 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