Daily Lesson Plan – Systems of Equations and Inequalities – Day 2
Aim(s):
Do Now:
Agenda:
SWBAT solve systems of equations NOT in slope intercept form by graphing.
SWBAT analyze special systems that have no solutions or an infinite number of solutions.
On the coordinate plane below, graph each of the following equations.
Be sure to label each equation a, b, c and d.
a.
b.
c.
d.
y = 3x – 3
y=2
y + 3 = 2x – 1
4x + y = 10
1. Do Now – 5 min.
5. Guided Practice – 5 min.
2. Problem of the Day – CASIO - 5 min.
6. Independent Practice - 20 min.
3. Blast from the Past – 5 min.
7. Exit Quiz – 8 min.
4. Introduction to New Material - 10 min.
8. Closing – 2 min.
1. Solve the following system by graphing and state the solution.
4x – y = -1
-x + y = x – 5
2. Solve the following system by graphing and state the solution.
Assessment:
2x – y = -5
2x – y = -1
3. Can a system of two linear equations have exactly two solutions? Explain.
POD:
4. Suppose you find that two linear equations are true when x = -2 and y = 3. What can you conclude about the
graphs of the equations?
Suppose your school is having a talent show to raise money for new band supplies. You think that 200 students
and 150 adults will attend. It will cost $1,000 to put on the talent show.
a. What is an equation that describes the ticket prices you can set for students and adults to raise $1,000?
b. What are three possible prices you could set for student and adult tickets?
Questions (1-4) find the slope of the line that is parallel to the graph of the equation.
1. y = x + 3
Blast from
the Past:
2. y =
−1
2
x−4
3. 3y + 2x = 7
4. 3x = 5y + 10
Questions (5-8) solve each equation for y.
5. 4x + 2y = 38
6.
1
2
x+
1
3
y=5
Key Ideas
7.
3
2
y=
4
5
x
8. 3x = 5y + 10
Potential Misunderstandings/Clarification
1. Systems of linear equations can be used to model problems.
2. Systems of equations can be solved in more than one way.
3. One method is to graph each equation and find the point of
intersection, if one exists.
In solving systems graphically, students may mistake a system
with infinite solutions for a system with no solutions.
Clarification:
A. Infinite solutions occur when two equations have the same
graph. All point that lie on the line are solutions to the system.
B. If the graphs of the equations are parallel, then the system
has no solutions because parallel lines will never intersect.
Vocabulary
Consistent – A system of equations that
has at least one solution is consistent.
Inconsistent – A system of equations that
has no solution is inconsistent.
Dependent – A consistent system that is
dependent has infinitely many solutions.
Independent – A consistent system that is
independent has exactly one solution.
Solution of a System of Linear
Equations – Any ordered pair that makes
all of the equations in a system true is a
solution of a system of linear equations.
Hook
Let’s return to yesterday’s discussion where we compared two cell phone plans. Yesterday we looked at examples where two
different plans had different starting costs as well as different monthly charges. In each case, we saw that eventually, if you waited
enough months, one of those plans became a better deal than the other one. Today, I want you to think about the case where two
plans have different starting costs, but the same monthly charge. Under this scenario, which do you think would be the better plan
to use?
Take one minute to turn and talk with your shoulder partner. How could you determine which cell phone plan will be the better
deal?
Introduction to New Material
A. Yesterday we learned how to represent the solution of two equations graphically. Take a moment to review what we did by
graphing and solving the following system:
y=x+2
y = 3x – 2
Check to make sure the solution you found makes both equations true.
B. All of the examples that we looked at yesterday were given to you in slope-intercept form, but of course things will not always be
quite this convenient. You reviewed solving for y in today’s “Blast from the Past” and at times you will have to apply this same skill
when graphing systems of equations. Let’s solve the following system by graphing, rewriting each equation in slope intercept form
first.
2x – y = -5
-2x – y = -1
What are we looking to have happen if we isolate y? We want y by itself on one side of the equation. Great. Let’s look just at the
first equation. How can we go about isolating y? We can subtract 2x from both sides of the equation. Okay, so what would that give
us? -y = -5 – 2x. Okay good, so am I done? Is y isolated? Yes, y is isolated, it is all by itself on the left side of the equation. Really? Is
it just y that is all by itself? Well no, it is –y. Oh okay, so I am going to ask again, is y all by itself on one side of the equation? No, we
have –y and we need this to be written so that just y is isolated. Okay good, we’re on the right track now. What can I do to both
sides of the equation in order to make the negative y into a positive y? If you divide each side of the equation by -1, you will be left
with y on the left side of the equation. Great, so what will this equation be now that y and only y has officially been isolated?
y = 5 + 2x. Great, now rewrite the second equation so that y is isolated on one side of the equation.
C. Now try graphing the following system. Remember to express both equations in slope intercept form first.
y + 6 = 5x
y = 5x + 4
What does your graph look like? Two parallel lines with a slope of 5 and y-intercepts of -6 and 4.
Right. Take 1 minute to check in with your should partner and decide what you think the solution of this system of equations is?
There isn’t a solution because the graphs never intersect. So when each equation in a system has the same slope, what can we say
about their solution? When two equations have the same slope, it means that they are parallel lines. We know that parallel lines
never intersect so when this happens, the system does not have any solutions.
D. Now try graphing the following system:
3y = x – 2
-6y = -2x + 4
What does your graph look like? The equations represent the same line. Yes – did anyone notice this was going to happen before
they started to graph it? A few hands go up. How did you know? When I expressed the equations in slope intercept form, I saw that
they had the same slope and the same y-intercept. Great – sometimes this happens. Sometimes we graph each equation and find
that they actually represent the same line. Take 1 minute to check in with your shoulder partner and decide what you think the
solution to this system of equation is. Every point on the line is a solution to the system. Right – so how could we quantify this?
Does this system have one solution, no solutions, what do you think? Since every point on the line is a solution, and there are
infinitely many points that can be graphed on the line, the system has infinitely many solutions.
E. NOTES
One Solution
No Solution
Infinitely Many Solutions
Description
The lines intersect at one point. The
lines have different slopes.
The lines are parallel. The lines have the
same slope and different y-intercepts.
The lines are the same. The
lines have the same slope and
y-intercept.
Vocabulary
The equations are consistent and
independent.
The equations are inconsistent.
The equations are consistent
and dependent.
Graph
Guided Practice
Woven into “Introduction to New Material Above.” Before moving onto independent practice, pose the following questions to the
group. Students should conference with their shoulder partner and then share out to the class.
1. Before we can graph the following system of equations, what do we need to do? What is the slope and y-intercept of each of the
following equations?
4x – y = -1
-x + y = x – 5
2. If you were asked to solve the following system of equations, what would you say about their solution(s)?
y = 3x + 2
3x – 4 = y
I can tell right away that this system does not have any solutions. Both equations have a slope of 3 which means that they are
parallel. Since parallel lines never intersect, these graphs never intersect. Because the graphs never intersect, the system has no
solutions.
3. How many solutions does the following system of equations have?
3x + 2 = y
y = 3x + 2
These two equations represent the same line. Because they are the same line, they have an infinite number of points of intersection.
Because they have an infinite number of points of intersection, the system has an infinite number of solutions.
4. What is an example of an inconsistent system of linear equations written in slope-intercept form? Students should present two
equations with the same slopes, but different y-intercepts.
Independent Practice You do (and do and do)…
Pearson, Algebra 1 Textbook (pages 363-365):
Solve each system by graphing. Tell whether the system has one solution, infinitely many solutions, or no solution.
1. y = 2x
y = -2x + 8
2. y = x + 3
y=x–1
3. y = 2x – 1
3y = 6x – 5
4. 3x + y = 2
4y = 12 – 12x
5. 2x – 2y = 5
y=x–5
6. 2y = x - 2
3
3y = x – 3
7. 3x – y = 2
4y = -x + 5
8. y – x = 5
3y = 3x + 15
2
9. A student graphs the system y = -x + 3 and y = -2x – 1. The graph that they create does not seem to have a point of intersection.
The student concludes that there is no solution. Describe and correct the student’s error.
10. Suppose you graph a system of linear equations and the intersection point appears to be (3, 7). Can you be sure that the
ordered pair (3, 7) is the solution? What must you do to be sure?
Closing Reinforce the key takeaways and connect to past and future learning.
To close out, let’s return to today’s cell phone scenario. Imagine that you have two cell phone plans. There are two things that we
can adjust with these plans, either the initial charge for the plan or the monthly charge. Is there ever a situation where one plan will
ALWAYS be a better deal than the other, regardless of how many months you have owned the cell phone?
Yes, the case in which one plan has a lower starting cost, but the same monthly charge as the other plan. If we were to graph this
scenario, we would have two parallel lines that will never intersect.
Is it possible to have a cell phone plan that will ALWAYS be the same cost as the other one?
Yes, if the initial charge and the monthly charge are the same, the graphs will be the same and one plan will never be a better deal
than the other one.