Solving Systems of Equations by Graphing Lesson Plan
Class and grade level: Algebra 1B, 10th
Period: 3
Knowledge: Big Ideas
An equation is a symbolic representation of all its possible solutions. A T-chart is a numerical
representation of a set of the possible solutions. A graph depicts a line, which is a representation of the
possible solutions. Any point on the line is a possible solution to the equation. Representing a set of
information numerically, symbolically, or graphically depicts the same solutions. Two or more equations
is called a system. To solve a system is to find the possible solutions that work for all of the equations in
the system. Numerically it would be difficult to solve this problem as only a portion of total possible
solutions are displayed. Symbolically solving a system of equations would require manipulation and
rearrangement of the equations in the system. Graphically representing the system’s equations
provides a quick, visual method to solving systems. A line represents all the possible solutions of a
particular equation. Graphing two equations simultaneously provides the possible solutions of both
equations. In the event of an intersection, the equations share one common solution – the point of
intersection. This point, or ordered pair, represents the solution to the system – an ordered pair that
works for all of the equations in the system. In the event of parallel lines, the equations do not have any
common solutions, and therefore there is no solution to the system. In the event that the system
contains two equations that are the same line, the two equations contain all of the same possible
solutions, and therefore the system has an infinite number of solutions.
Knowledge: Experiences, Patterns, and Explanations
Observations or
experiences (examples,
phenomena, data)
Graphs of :

Y = 2X – 1

Y=X+1

Y=5

Y = 3X + 6

X=7
Graph simultaneously:

Y = 2X – 1

Y=X+1


2X + 2Y = -10
-X + Y = -3


4X + 2Y = 22
Y=5
Graph simultaneously:

Y = 2X – 1

Y = 2X + 1


Y = 3X + 4
Y = 3X - 7


2X + 3Y = 6
4X + 6 Y = -12
Patterns (laws,
generalizations, graphs,
tables, categories)
Explanations (models,
theories)
The graph of each equation is a line
representing all of the possible
solutions for that particular
equation.
An equation is a visual representation of
graphing the infinite number of
solutions. A particular input (X),
produces a specific output (Y).
Each pair of equations represents
lines with different slopes.
Each pair of equations produces a
graph with two lines which intersect
at one specific point (an ordered
pair).
Each pair of equations represents
lines with the same slopes.
Each pair of equations produces a
graph with two parallel lines.
The point at which the lines of two
equations intersect is the solution to the
system of equations.
There are no points of intersection
between parallel lines and therefore
there are no solutions to the system of
equations.
Graph simultaneously:

Y = 3X + 1

2Y = 6X + 2


3X + 2Y = 6
6X + 4Y = 12


-2X + 3Y = 9
4X – 6Y = -18
Each pair of equations represents
lines with the same slope.
Each pair of equations are multiples
of each other.
The lines are the same line so there are
an infinite number of points of
intersection. Therefore there are many
solutions to the system of equations.
Each pair of equations produces a
graph in which the lines are
superimposed on each other.
Application: Model-based Reasoning
Inquiry: Finding and Explaining Patterns in Experience
Practices: Objectives for Student Learning
Objective
Michigan Objectives
1. Analyze, interpret, and translate among representations of patterns, including tables, charts,
graphs, matrices, and vectors
Specific Topic Objectives
1. Give 3 representations of the same pattern (numerical, graphical, symbolic)
2. Construct a graph and write an algebraic equation that describes the graph from a table of data.
1)
2)
3)
4)
As a class review how to graph a line and what the line means. 10 minutes
Write systems of equations on the board.
Divide the class into three groups.
Have each group solve one of each kind of system by graphing. (1 solution, no solution,
infinite solutions) 10 minutes
5) Have a representative from each group draw their answers on the board. 5 minutes
6) Ask each group to look for patterns between the different systems based upon classifying
the systems by number of solutions.
7) Write the patterns on the board (class notes).
8) Come up with explanations for the patterns. (class notes)
9) To solve problems determine (1) slope of each line and (2) if equations are multiples of each
other. Then graph them and determine the solutions.
1 solution: different slopes solution is point of intersection
no solution: same slope  parallel lines  no solution
infinite solutions: same slope & multiples of each other same line infinite solutions
10) Example problems
11) Pass out homework  due Friday at the beginning of the hour.
Due date:____________
Name:______________
Solving Systems of Equations By Graphing
Please solve each system by graphing. Show your graphs and your solutions.
1) Y = 3X + 2
Y = 2X + 1
4) Y = 5X + 3
2Y = 10X + 6
7) X + Y = 4
2X + Y = 2
2) Y = 2X + 5
-2X + Y = 8
5) 2X + 2Y = 6
Y = -X + 4
8) Y = X + 1
Y–X=1
3) 4Y = 2X + 8
-½ X = 2 - Y
6) 2Y = 2X + 4
Y = 2X + 4
9) Y = ¾ X + 2
- 3X + 4Y = 8
10) X + 2Y = 1
4Y= -2X + 4
11) 4Y = 4X + 16
Y = 2X + 3
12) 2X + 2Y = 6
Y = -X + 3
13) Y = 3X + 2
-3X -2 = -Y
14) Y = 2X + 2
Y = 2X - 2
15) Y = 4X - 1
2X + Y = 1
16) X + Y = 2
5Y = -5X + 10
17) Y = ¼ X - 3
12 – X = - 4Y
18) Y= 3X + 2
-3X= 2 - Y