Download Solving Systems of Equations using Substitution

DRILL:
1.
2x + 4y =1
x - 4y =5
2. 2x – y =6
x+y=3
Using Elimination to Solve a
Word Problem:
Two angles are supplementary. The
measure of one angle is 10 degrees
more than three times the other.
Find the measure of each angle.
Using Elimination to Solve a
Word Problem:
Two angles are supplementary. The
measure of one angle is 10 more
than three times the other. Find the
measure of each angle.
x = degree measure of angle #1
y = degree measure of angle #2
Therefore x + y = 180
Using Elimination to Solve a
Word Problem:
Two angles are supplementary. The
measure of one angle is 10 more
than three times the other. Find the
measure of each angle.
x + y = 180
x =10 + 3y
Using Elimination to Solve a
Word Problem:
Solve
x + y = 180
x =10 + 3y
x + y = 180
-(x - 3y = 10)
4y =170
y = 42.5
x + 42.5 = 180
x = 180 - 42.5
x = 137.5
(137.5, 42.5)
Using Elimination to Solve a
Word Problem:
The sum of two numbers is 70
and their difference is 24. Find
the two numbers.
Using Elimination to Solve a
Word problem:
The sum of two numbers is 70
and their difference is 24. Find
the two numbers.
x = first number
y = second number
Therefore, x + y = 70
Using Elimination to Solve a
Word Problem:
The sum of two numbers is 70
and their difference is 24. Find
the two numbers.
x + y = 70
x – y = 24
Using Elimination to Solve a
Word Problem:
x + y =70
x - y = 24
2x = 94
x = 47
47 + y = 70
y = 70 – 47
y = 23
(47, 23)
Now you Try to Solve These
Problems Using Elimination.
Solve
1. Find two numbers whose sum is
18 and whose difference is 22.
2. The sum of two numbers is 128
and their difference is 114. Find
the numbers.
3.4 Systems of Linear Inequalities
Objectives:
•Write and graph a system of linear inequalities in two variables
•Write a system of linear inequalities in two variables for a given
solution region
Example 1
Graph the system.
x  0
y  0


 y  3x  2

 y  x  4
8
6
4
2
-8 -6 -4 -2
-2
-4
-6
-8
2 4 6 8
Example 1
Graph the system.
x  0
y  0


 y  3x  2

 y  x  4
8
6
4
2
-8 -6 -4 -2
-2
-4
-6
-8
2 4 6 8
Example 2
Graph the system.
 y  2x  2

1

y  x  1
2


 x  3
8
6
4
2
-8 -6 -4 -2
-2
-4
-6
-8
2 4 6 8
Example 2
Graph the system.
 y  2x  2

1

y

x 1

2


 x  3
8
6
4
2
-8 -6 -4 -2
-2
-4
-6
-8
2 4 6 8
Practice
Graph the system.
1) x  0
y  0


 y  x  2
 y  2x  3
Example 3
Write the system of inequalities graphed below.
x<3

 y  2

 y  x  6

 y < 2x + 6
8
6
4
2
-8 -6 -4 -2
-2
-4
-6
-8
2 4 6 8
Example 4
Graph the linear inequality.
5  x  3
x3
x  5
8
6
4
2
-8 -6 -4 -2
-2
-4
-6
-8
2 4 6 8
Example 4
Graph the linear inequality.
5  x  3
x3
x  5
8
6
4
2
-8 -6 -4 -2
-2
-4
-6
-8
2 4 6 8
Example 5
Graph the linear inequality.
4  y  2
y2
y  4
8
6
4
2
-8 -6 -4 -2
-2
-4
-6
-8
2 4 6 8
Example 5
Graph the linear inequality.
4  y  2
y2
y  4
8
6
4
2
-8 -6 -4 -2
-2
-4
-6
-8
2 4 6 8
Warm-Up
Solve using substitution.
5y  20

x  3y  4z  11
 y  2z  2

Example 1
Use elimination to solve the system.
8x  2y  10

4x  15  3y
8x + 2y = -10
(-2) (4x – 3y) =(15) (-2)
8x + 2y = -10
-8x + 6y = -30
8y = -40
y = -5
write in standard form
multiply as needed
addition property
8x + 2(-5) = -10
8x - 10 = -10
8x = 0
x=0
(0,-5)
Example 2
Use elimination to solve the system.
5x  3y  12

5x  3y  15
5x + 3y = 12
(-1) (5x + 3y )= (15) (-1)
5x + 3y = 12
-5x – 3y = -15
write in standard form
multiply as needed
addition property
0 = -3
huh?
there is no solution
the system is inconsistent
Example 3
Use elimination to solve the system.
16x  9y  8

4x  2.25y  2
16x – 9y = 8
(-4) (4x – 2.25y) =(2) (-4)
16x - 9y = 8
-16x + 9y = -8
write in standard form
multiply as needed
addition property
0=0
true
consistent, dependent
solution is all points on the
graph of either equation
Practice
Use elimination to solve each system.
1) 4x  4y  8
x  4y  12
2) 2x  3y  5

x  2y  4