1. U2 Day 1 Systems of Equations

2. Math3H – U2 Day 1 Systems of Equations

3. Unit 2 Day 1 - OBJECTIVES To understand what a system of equations is.
Be able to solve a system of equations using graphing, substitution, and elimination.
Determine whether the system has one, none, or infinite solutions.
Be able to graph equations without using a graphing calculator.

4. 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

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
Point of intersection 2 1 2

6. 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

7. ax + by = c Standard Form of a Linear Equation
NOTE: The equation ax + by = c is just
another form of a linear equation.
TO GRAPH: Change it to y = mx + b
OR use the x and y-intercepts 

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

9. Solutions of Systems No Solution: when lines of a graph are parallel
since they do not intersect, there is no solution
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.

10. Solutions of Systems Infinite Solutions:
a pair of equations that have the same slope and y-intercept.
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.

11. Solutions of Systems One Solution:
the lines of two equations intersect
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. Examples… How many solutions?
Look at the slope and y-intercepts in your group 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

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

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

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)

16. 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 over 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.

17. Graphing Manually
Plot two points by finding the x and y-intercepts …Substitute 0 for x and then y!
2x + 3y = 4

18. Graphing Manually
Plot two points by finding the x and y-intercepts …Substitute 0 for x and then y!
2x + 3y = 4
1. Substitute 0 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)

19. Graphing Manually
Plot two points by finding the x and y-intercepts …Substitute 0 for x and then y!
2x + 3y = 4
3. Substitute 0 for y, then solve for x
y = 0
2x + 3(0) = 4
2x = 4
x = 2
4. Plot the point: (2, 0)

20. Graphing Manually cont.
Plot two points by finding the x and y-intercepts …Substitute 0 for x and then y!
5. Draw line connecting both points.

21. More Examples
HOW MANY SOLUTIONS?? If “one solution” graph it and give the point of intersection. NON-Calculator! 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

22. SO… What do we know? 
You can graph systems of equations from standard form and slope-intercept form.
You can determine how many solutions there are: none/one/infinite
Sometimes you can tell how many solutions without graphing if you look at the slopes and intercepts!

23. HOMEWORK 
U2 Day 1 HW