SOLVING SYSTEMS OF EQUATIONS

AUDIENCE: Algebra 1
TIME FRAME: 10 40-minute period.

NYS PERFORMANCE STANDARDS:
AA7 Analyze and solve verbal problems whose solution requires solving systems of
linear equations in two variables .
AA10 Solve systems of two linear equations in two variables algebraically.
AG7 Graph and solve systems of linear equations and inequalities with rational
coefficients in two variables .
ACM10 Solve systems of linear equations using concrete models, graphs, tables,
algebraic methods and check the reasonableness of solutions.

NCTM STANDARDS:
• Understand vectors and matrices as systems that have some of the properties of
the real-number system
• Develop an understanding of properties of, and representations for, the addition
and multiplication of vectors and matrices
• Develop fluency in ope rations with real numbers, vectors, and matrices, using
mental computation or paper and pencil calculations for simple cases and
technology for more complicated cases.
• Judge the reasonableness of numerical computations and their results.

OBJECTIVES:
By the end of this unit, students will be able to:
1. Solve systems of equations using various methods including substitution ,
elimination using addition and subtraction , and elimination using multiplication,
2. Use a graphing calculator to solve systems of equations by graphing and matrices.

RESOURCES:
Holliday, Cuevas, Moore-Harris, Carter, Marks, Casey, Day, Hayek, Algebra 1; New
York State Edition, McGraw Hill, Glencoe, New York, 2006.

OVERVIEW

*NOTE: Days can be rearranged; only elimination by addition should be taught before
elimination by multiplication.

Use Algebra Ch 5 APP from TI at any point suitable.
This app reviews consistent and independent, consistent and dependent, and inconsistent.
It also explains the elimination method and has a couple of games to help students master
the subject.

ANTICIPATORY SET:
Marcus went to the movies and bought 2 tubs of popcorn and 1 soda for $4.75. Jackie
went to the movies and bought 1 tub of popcorn and 2 sodas for $4.25. What is the cost
of 1 soda and 1 popcorn?

DAY 1 – GRAPHING
Objectives
Solving systems by graphing.

ATTACHED: Graph paper, graphing activity, graphic organizer to help with
terminology.

Students should be familiar with graphing a line on the graphing calculator. If they are
not, they need to know how to input a linear equation and how to read the table to plot
the ordered pairs .

Key Terms
In general, a solution of a system in two variables is an ordered pair that makes
BOTH equations true.
In other words, it is where the two graphs intersect, what they have in common. So if an
ordered pair is a solution to one equation, but not the other, then it is NOT a solution to the
system.
A consistent system is a system that has at least one solution.
An inconsistent system is a system that has no solution; parallel lines.
The equations of a system are dependent if ALL the solutions of one equation are also
solutions of the other equation. In other words, they end up being the same line.
The equations of a system are independent if they do not share ALL solutions. They
can have one point in common, just not all of them.

Use green globs to show examples of the different types of systems and work on
vocabulary. Graph the fol lowing systems then discuss the solutions and the terminology in
the chart below :
(*note – these are not in y=mx+b format as Green Globs does not require this format –
decide whether this is a good time to work on solving for y or to work on the vocabulary;
students need to be accustomed to seeing the equations in different forms and know that
they can still solve them!)

1) x+2y=3; 3x-y=-5, 1 solution, consistent and dependent.
2) y=2x-3; 4x=2y+6, Infinite solutions, consistent and dependent.
3) 3x-y=-2; 3x-y=0, no solution, inconsistent; lines are parallel.

Have students sketch the three different examples in their notebooks and label each with the
appropriate vocabulary. The note should resemble the chart below.

	INTERSECTING LINES	SAME LINE	PARALLEL LINES
GRAPH
# OF SOLUTIONS	One Solution	Infinite Solutions	No Solution
TERMINOLOGY	Consistent Independent	Consistent Dependent	Inconsistent

If the above websites are not used, the following should be completed.

Using the graphing calculator, the equations must be in y=mx+b form. In Y= type in
both equations. Hit GRAPH to show the lines. To find the point of intersection, the
point needs to be on the viewing window. This may require adjusting the window or
hitting zoom, fit. Then, find the intersection by 2^nd Trace (Calc), choose 5 Intersection.
Follow the prompts on the screen, the calculator should move from curve to curve as
enter is hit but it may not if it is close to the side of the window, it may need manually
moved from curve to curve by using the right and left arrows. After choosing curve 1,
curve 2, and guess, the point of intersection will be shown.
The table (2^nd Graph) can also be used if the solution(s) is a whole number. The table can
reset (2^nd window, change in TBL ) the change in values to view decimal solutions ,
however, this gets tricky for a level 1 algebra course. The solution is where the x value
has two of the same y values. For example if x is 2 and y1 is 5 and y2 is 5, the solution is
(2, 5). Students should try to estimate the answer before going to the table as it will make
the solution easier to locate.

Use the smartview to show the students the steps on the overhead. Do examples that the
window will need to be adjusted. If students have a difficult time adjusting the window,
they can use zoom fit. Zoom standard is also a good key for them to remember as it
brings back the standard viewing window.

Ex. 1
Y=-6; 4x+y=2
- solve both equations for y
- graph & find intersection (solution)
- describe the solution

Ans – y=-6, y=-4x+2; (2, -6); one solution, consistent and independent.

Ex. 2
X+y=24; 3x+y=44
- solve both equations for y
- graph & find intersection (adjust window; xmax 15, ymax 50 or zoom fit)
- describe the solution

Ans – y=-x+24, y=-3x+44; (10, 14); one solution, consistent and independent

Pass out graph paper (attached below) and have students graph, find the solutions, and
describe the solutions to the following systems.

1. 2x-y=-3; 8x-4y=-12; Ans - infinite solutions, consistent, dependent
2. x-2y=4; x-2y=-2; Ans -no solution, inconsistent
3. x+y=3; 12x+4y=20; Ans – (1,2), one solution, consistent, independent