1. 3-1 Solving Systems
Using Tables and
Graphs
Algebra II Unit 3 Linear Systems
© Tentinger

2. Essential Understanding and
Objectives
• Essential Understanding: to solve a system of equations, find a set of
values that replace the variables in the equations and make each
equation true.
• Objectives:
• Students will be able to solve linear system using a graph or table

3. Iowa Core Curriculum
• Algebra
• A. CED.2 Create equations in two or more variables to represent
relationships between quantities; graph equations on coordinate axes
with labels and scales.
• A.CED.3 . Represent constraints by equations or inequalities, and by
systems of equations and/or inequalities, and interpret solutions as
viable or nonviable options in a modeling context. For
example, represent inequalities describing nutritional and cost
constraints on combinations of different foods.
• A.REI.6 Solve systems of linear equations exactly and approximately
(e.g., with graphs), focusing on pairs of linear equations in two variables.
• A.REI.11 Explain why the x-coordinates of the points where the graphs
of the equations y = f(x) and y = g(x) intersect are the solutions of the
equation f(x) = g(x); find the solutions approximately, e.g., using
technology to graph the functions, make tables of values, or find
successive approximations. Include cases where f(x) and/or g(x) are
linear, polynomial, rational, absolute value, exponential, and logarithmic
functions.★

4. System of Equations
• System of equations: when you have two or more related
unknowns, you can represent them with a set of two or more
equations.
• Linear System: consists of linear equations
• Solution of a System: the set of values for the variables that makes
all the equations true.
• There could be one solution
• Infinitely many solutions
• No Solution
• You can use a graph or table to solve a system of equations

5. What is the solution?
• Two different methods to solve:
• Graphing
• Table
ì y = -x + 3
ì x - 2y = 4 ï
í í 3
î3x + y = 5 ïy = x - 2
î 2
ì3x + y = 5
í
îx - y = 7

6. Example
• If the growth rates
• Greenland Shark continue, how long will
Growth Rate: 0.75 cm/yr each shark be when it
Birth Length: 37 cm is 25 years old?
• Spiny Dogfish Shark • Explain why growth
Growth Rate: 1.5 cm/yr rates for these sharks
Birth Length: 22 cm may not continue
indefinitely.
• If the growth rates stayed
the same at what age
would the two sharks be
the same length?

7. Example
• The table shows the populations of San Diego and Detroit.
• Assuming the trends continue, when will the population of these
cities be equal? What will the population be?
• Step 1: Enter the Data into your lists on your calc
• L1: Number of Years since 1950
• L2: San Diego population
• L3: Detroit Population
• Step 2: In the Stat Plot, turn on plots 1 and 2. In Plot 2 change list L2
to L3 by using the 2nd 3 (L3) to get the L3 list. Adjust window
1950 1960 1970 1980 1990 2000
San Diego 334,387 573,224 696,769 875,538 1,110,549 1,223,400
Detroit 1,849,568 1,670,144 1,511,482 1,203,339 1,027,974 951,270

8. Example
• The table shows the populations of San Diego and Detroit.
• Assuming the trends continue, when will the population of
these cities be equal? What will the population be?
• Step 3: Calc the LinReg for both populations (you will need to
type information into L2)
• San Diego:17816x + 356896
• Detroit: -19217x + 1849401
• Step 4: Calculate intersection of the lines
• Answer: 1990 and about 1,074,917
1950 1960 1970 1980 1990 2000
San Diego 334,387 573,224 696,769 875,538 1,110,549 1,223,400
Detroit 1,849,568 1,670,144 1,511,482 1,203,339 1,027,974 951,270

9. Classifying a System of two
Linear Equations
• Independent System: has one solution
• Independent
• 2 lines intersect at one point
• the slopes of the lines are not equal
• the y-intercepts may or may not be equal
• Dependent System: has infinitely many solutions
• Dependent
• 2 lines coincide
• the slopes of the lines are equal
• the y-intercepts are equal
• Inconsistent System: No Solution
• Inconsistent
• 2 parallel lines
• the slopes of the lines are equal
• the y-intercepts are not equal

10. Example
• Without graphing, is the system independent, dependent, or
inconsistent?
ì-3x + y = 4
ï
í 1
ïx - y = 1
î 3 ì2x + 3y = 1
í
î 4x + y = -3
ì y = 2x - 3
í
î6x - 3y = 9

11. Homework
• Pg. 138 – 139
• #2 – 40 even
• 19 problems