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Algebra 2
Solving Systems Using Tables and
Graphs
Lesson 3-1
Goals
Goal
• To solve a linear system
using a graph or a table.
Rubric
Level 1 – Know the goals.
Level 2 – Fully understand the
goals.
Level 3 – Use the goals to
solve simple problems.
Level 4 – Use the goals to
solve more advanced problems.
Level 5 – Adapts and applies
the goals to different and more
complex problems.
Vocabulary
• System of Equations
• Linear System
• Solution of a System
Definitions
• System of equations - is a set of two or more
equations containing two or more variables.
– Example:  x 2  2 x  y  7  0 Equation 1

3 x  y  1  0
Equation 2
• Linear system - is a system of equations
containing only linear equations.
– Example: a)  Ax  By  C Equation 1

Dx  Ey  F Equation 2
2x  y  9
b) 

3x  4y  8
Definitions
• Solution of a System - Recall that a
line is an infinite set of points that
are solutions to a linear equation.
The solution of a system of
equations is the set of all points that
makes all the equations true.
• On the graph of the system of two
equations, the solution is the set of
points where the lines intersect. A
point is a solution to a system of
equations if the x- and y-values of
the point satisfy both equations.
Example:
Use substitution to determine if the given ordered pair is an
element of the solution set for the system of equations.
x – 3y = –8
(1, 3);
3x + 2y = 9
x – 3y = –8
3x + 2y = 9
(1) –3(3) –8
3(1) +2(3) 9
–8
–8 
Substitute 1 for x and 3
for y in each equation.
9
9
Because the point is a solution for both equations, it is a solution of
the system.
Example:
Use substitution to determine if the given ordered pair is an
element of the solution set for the system of equations.
(–4,
x + 6 = 4y
);
2x + 8y = 1
x + 6 = 4y
(–4) + 6
2
2
2x + 8y = 1
Substitute –4 for x and
for y in each equation.
2(–4) +
1
–4
Because the point is not a solution for both equations, it is not a
solution of the system.
1x
Your Turn:
Use substitution to determine if the given ordered pair is an
element of the solution set for the system of equations.
6x – 7y = 1
(5, 3);
3x + 7y = 5
6x – 7y = 1
3x + 7y = 5
6(5) – 7(3) 1
9
1x
3(5) + 7(3)
Substitute 5 for x and 3
for y in each equation.
36
5
5x
Because the point is not a solution for both equations, it is not a
solution of the system.
Your Turn:
Use substitution to determine if the given ordered pair is an
element of the solution set for the system of equations.
x + 2y = 10
(4, 3);
3x – y = 9
3x – y = 9
x + 2y = 10
(4) + 2(3) 10
10
10 
Substitute 4 for x and 3
for y in each equation.
3(4) – (3)
9
9
9 
Because the point is a solution for both equations, it is a solution of
the system.
Your Turn:
Use substitution to determine if the given ordered pair is an
element of the solution set of the system of equations.
x+y=2
1. (4, –2)
x + 3y = –9
2. (–3, –2)
y – 2x = 4
y + 2x = 5
no
yes
Graphs & Tables
• Just as you can use graphs or tables to find
some of the solutions to a linear equation.
You can do the same to find solutions to
linear systems.
How to Use Graphs to Solve
Linear Systems
Consider the following system:
x – y = –1
y
x + 2y = 5
We must ALWAYS verify that your
coordinates actually satisfy both
equations.
(1 , 2)
x
To do this, we substitute the
coordinate (1 , 2) into both
equations.
x – y = –1
(1) – (2) = –1 
x + 2y = 5
(1) + 2(2) =
1+4=5
Since (1 , 2) makes both equations
true, then (1 , 2) is the solution to the
system of linear equations.
Graphing to Solve a Linear
System
Solve the following system by graphing:
3x + 6y = 15
–2x + 3y = –3
Start with 3x + 6y = 15
Subtracting 3x from both sides yields
6y = –3x + 15
While there are many different
ways to graph these equations, we
will be using the slope - intercept
form.
To put the equations in slope
intercept form, we must solve both
equations for y.
Dividing everything by 6 gives us…
1
2
y= -
x+
5
2
Similarly, we can add 2x to both
sides and then divide everything by
3 in the second equation to get
y=
2
3
Now, we must graph these two equations.
x- 1
Graphing to Solve a Linear
System
Solve the following system by graphing:
y
3x + 6y = 15
–2x + 3y = –3
Using the slope intercept form of these
equations, we can graph them carefully
on graph paper.
y = - 12 x +
y = 23 x - 1
x
(3 , 1)
5
2
Start at the y - intercept, then use the slope.
Label the
solution!
Lastly, we need to verify our solution is correct, by substituting (3 , 1).
Since 3(3)+ 6 (1) = 15 and - 2(3)+ 3(1) = - 3, then our solution is correct!
Graphing to Solve a Linear
System
Step 1: Put both equations in slope intercept form.
Solve both equations for y, so that
each equation looks like
y = mx + b.
Step 2: Graph both equations on the
same coordinate plane.
Use the slope and y - intercept for
each equation in step 1. Be sure to
use a ruler and graph paper!
Step 3: Find the point where the
graphs intersect.
This is the solution! LABEL the
solution!
Step 4: Check to make sure your
solution makes both equations true.
Substitute the x and y values into both
equations to verify the point is a
solution to both equations.
Graphing to Solve a Linear
System
Solve the following system of equations by graphing.
2x + 2y = 3
x – 4y = -1
Step 1: Put both equations in slope intercept form.
y = - x + 32
y = 14 x + 14
y
LABEL the solution!
(1, 12 )
x
Step 2: Graph both equations on the
same coordinate plane.
Step 3: Estimate where the graphs
intersect. LABEL the solution!
Step 4: Check to make sure your
solution makes both equations true.
2(1)+ 2(12 )= 2 + 1 = 3
1- 4(12 ) = 1- 2 = - 1
Example:
Use a graph and a table to solve the system. Check your
answer.
2x – 3y = 3
y+2=x
Solve each equation for y.
y=
x–1
y= x – 2
Example: continued
y=
x–1
y= x – 2
On the graph, the lines appear
to intersect at the ordered pair
(3, 1)
Example: continued
Make a table of values
for each equation. Notice
that when x = 3, the yvalue for both equations
is 1.
The solution to the
system is (3, 1).
y=
x–1
x
0
y
–1
y= x – 2
x
y
0
–2
1
1
–1
2
2
0
3
1
3
1
Example: Graphing Calculator
Use a graph and a table to solve the system. Check your
answer.
x–y=2
2y – 3x = –1
Solve each equation for y.
y=x–2
y=
Example: Graphing Calculator
Use your graphing calculator to graph
the equations and make a table of
values. The lines appear to intersect at
(–3, –5). This is the confirmed by the
tables of values.
The solution to the system is (–3, –5).
Check Substitute (–3, –5) in the original
equations to verify the solution.
x–y = 2
(–3) – (–5)
2
2y – 3x = –1
2
2
2(–5) – 3(–3) –1

–1
–1 
Your Turn:
Use a graph and a table to solve the system. Check your
answer.
2y + 6 = x
4x = 3 + y
y= x – 3
Solve each equation for y.
y= 4x – 3
Your Turn: continued
y= x – 3
y= 4x – 3
On the graph, the lines
appear to intersect at the
ordered pair (0, –3)
Your Turn: continued
y= x–3
y = 4x – 3
Make a table of values for
each equation. Notice that
when x = 0, the y-value for
both equations is –3.
x
y
x
y
0
–3
0
–3
1
1
The solution to the
system is (0, –3).
2
2
5
3
9
1
3
–2
Your Turn:
Use a graph and a table to solve the system. Check your
answer.
x+y=8
2x – y = 4
Solve each equation for y.
y=8–x
y = 2x – 4
Your Turn: continued
y=8–x
y = 2x – 4
On the graph, the lines
appear to intersect at the
ordered pair (4, 4).
Your Turn: continued
y= 8 – x
Make a table of values for
each equation. Notice that
when x = 4, the y-value for
both equations is 4.
The solution to the system is
(4, 4).
y = 2x – 4
x
y
x
y
1
7
1
–2
2
6
2
0
3
5
3
2
4
4
4
4
Your Turn:
Use a graph and a table to solve each system. Check your
answer.
y–x=5
3x + y = 1
Solve each equation for y.
y= x + 5
y= –3x + 1
Your Turn: continued
y= x + 5
y= –3x + 1
On the graph, the lines
appear to intersect at the
ordered pair (–1, 4).
Your Turn: continued
y= x + 5
Make a table of values for each
equation. Notice that when
x = –1, the y-value for both
equations is 4.
The solution to the system is
(–1, 4).
y= –3x + 1
x
y
x
y
–1
4
–1
4
0
5
0
1
1
6
1
–2
2
7
2
–5
Example: Application
City Park Golf Course charges $20 to rent golf clubs plus $55
per hour for golf cart rental. Sea Vista Golf Course charges
$35 to rent clubs plus $45 per hour to rent a cart. For what
number of hours is the cost of renting clubs and a cart the
same for each course?
Example: continued
City Park Golf Course charges $20 to rent golf clubs plus $55 per hour
for golf cart rental. Sea Vista Golf Course charges $35 to rent clubs plus
$45 per hour to rent a cart. For what number of hours is the cost of
renting clubs and a cart the same for each course?
Step 1 Write an equation for the cost of renting clubs
and a cart at each golf course.
Let x represent the number of hours and y represent
the total cost in dollars.
City Park Golf Course: y = 55x + 20
Sea Vista Golf Course: y = 45x + 35
Because the slopes are different, the system is independent and
has exactly one solution.
Example: continued
Step 2 Solve the system by using a table of values.
Use increments of
represent 30 min.
When x =
, the yvalues are both
102.5. The cost of
renting clubs and
renting a cart for
hours is $102.50 at
either company. So
the cost is the same
at each golf course
for hours.
to y = 55x + 20
x
0
y
20
y = 45x + 35
x
0
57.5
47.5
1
75
1
120
80
102.5
102.5
2
y
35
2
125
Your Turn:
Ravi is comparing the costs of long distance calling cards. To
use card A, it costs $0.50 to connect and then $0.05 per minute.
To use card B, it costs $0.20 to connect and then $0.08 per
minute. For what number of minutes does it cost the same
amount to use each card for a single call?
Step 1 Write an equation for the cost for each of the different long
distance calling cards.
Let x represent the number of minutes and y represent
the total cost in dollars.
Card A: y = 0.05x + 0.50
Card B: y = 0.08x + 0.20
Your Turn: continued
Step 2 Solve the system by using a table of values.
When x = 10 , the y-values
are both 1.00. The cost of
using the phone cards of 10
minutes is $1.00 for either
cards. So the cost is the
same for each phone card
at 10 minutes.
y = 0.05x + 0.50 y = 0.08x + 0.20
x
y
x
y
1
0.55
1
0.28
5
0.75
5
0.60
10
1.00
10
1.00
15
1.25
15
1.40
Definitions
• The systems of equations in the past Examples have had
exactly one solution. However, linear systems may also
have infinitely many or no solutions.
• Consistent system - is a system of equations that has at
least one solution.
• Inconsistent system - is a system of equations that has no
solutions.
• You can classify linear systems by comparing the slopes
and y-intercepts of the equations.
• Independent system - has equations with different slopes.
• Dependent system - has equations with equal slopes and
equal y-intercepts.
Graphical Solutions of
Linear Systems
Example:
Classify the system and determine the number of solutions.
x = 2y + 6
3x – 6y = 18
y= x–3
Solve each equation for y.
y= x–3
The equations have
the same slope and
y-intercept and are
graphed as the same
line.
The system is consistent and dependent with infinitely many solutions.
Example:
Classify the system and determine the number of solutions.
4x + y = 1
y + 1 = –4x
y = –4x + 1
Solve each equation for y.
y = –4x – 1
The system is inconsistent and has no solution.
The equations have
the same slope but
different y-intercepts
and are graphed as
parallel lines.
Example: continued
Check A graph shows parallel lines.
Your Turn:
Classify the system and determine the number of solutions.
7x – y = –11
3y = 21x + 33
y = 7x + 11
Solve each equation for y.
y = 7x + 11
The equations have
the same slope and
y-intercept and are
graphed as the same
line.
The system is consistent and dependent with infinitely many
solutions.
Your Turn:
Classify each system and determine the number of solutions.
x+4=y
5y = 5x + 35
y=x+4
Solve each equation for y.
y=x+7
The system is inconsistent with no solution.
The equations have
the same slope but
different y-intercepts
and are graphed as
parallel lines.
Your Turn:
Classify each system and determine the number of
solutions.
y + 2x = –10
–4x = 2y – 10
5.
4.
y + 2x = –10
y - 2x = –1
consistent, independent; one
inconsistent; none
solution
Review
Your Turn:
What type of system of equations is
shown?
x+y=5
2x = y – 5
A. consistent and independent – one solution
B. consistent and dependent – infinite solutions
C. Inconsistent – no solution
D. none of the above
Your Turn:
What type of system of equations is
shown?
x+y=3
2x = –2y + 6
A. consistent and independent – one solution
B. consistent and dependent – infinite solutions
C. Inconsistent – no solution
D. none of the above
Your Turn:
What type of system of equations
is shown?
y = 3x + 2
–6x + 2y = 10
A. consistent and independent – one solution
B. consistent and dependent – infinite solutions
C. Inconsistent – no solution
D. none of the above
Your Turn:
Graph the system of equations below. Which statement is
not true?
f(x) = x + 2
g(x) = x + 4
A. f(x) and g(x) are consistent and
dependent – infinite solutions.
B. f(x) and g(x) are inconsistent – no
solution.
C. f(x) and h(x) are consistent and
independent – one solution.
D. g(x) and h(x) are consistent – at
least one solution.
Essential Question
Big Idea: Solving Equations and
Inequalities
• What does a solution of a system of linear
equations represent?
• A solution represents values that make both
equations true. To find the solution, graph the
equations and find the point where the lines
intersect. You can also make a table and find the
value of x that makes the y–values equal.
Assignment
• Section 3-1, Pg. 150 -153; #1 – 6 all, 8 -14
even, 18 – 50 even.