2. Introduction:
• Systems, 2 x 2 in this case, are when you have 2 equations and 2
unknowns (letters). Here's an example:
• x + y = 3
• x - 2y = 0
• The goal is to find an x guy and a y guy that work in both equations.
• I'll show you three different ways to solve these.
• The first method is graphing
• The second method is substitution
• The third method is elimination
3. Objectives:
•Solve systems of linear equations by:
1. Graphing
2. Substitution
3. Elimination
•Solve systems of inequalities
4. Definitions:
• System of equations : is a set or collection of equations that
you deal with all together at once
• System of inequalities: is a set of two or more inequalities
with the same variables
• Consistent: has a solution
1. Independent = 1 solution
2. Dependent = infinite solutions
• Inconsistent: no solution
5. How to solve a system of equations?
Y=2x+3
Y=11-2x
X Y=2X+3
0 2.0+3=3
1 2.1+3=5
0
1
Y=11-2x
11-2.0=11
11-2.1= 9
(0,3) (1,5)
(0,11) (1,9)
7. How to solve a system of equations?
Y=2x+3
Y=11-2x
2x+3=11-2x
2x+2x=11-3
4x=8
4x(/4)=8(/4)
X=2
Y=2.(2)+3
Y=4+3
Y=7
Y=11-2x
Y=11-2.(2)
Y=11-4
Y=7
8. Y=2x+3
Y=11-2x
How to solve a system of equations?
Y=2x+3
- Y=-2x+11
__________
0=4x-8
=4x-8
8=4x
8(/4)=4x(/4)
2=x
Y=2.(2)+3
Y=4+3
Y=7
Y=11-2x
Y=11-2.(2)
Y=11-4
Y=7
9. How to solve a system
of inequalities?
6x-3y<-9
-3y<-9-6x
Y<3+2x
4x-3y>=1
-3y>=1-4x
Y>=-0.33+1.33x
10. Real life application
• Two small pitchers and one large pitcher can hold 8 cups of
water. One large pitcher minus one small pitcher constitutes 2 cups
of water. How many cups of water can each pitcher hold?
Let x = small pitcher
y = large pitcher
2x + y = 8
y - x = 2
2x + y = 8
-x + y = 2
subtract:
3x = 6
x = 2
The small pitcher holds 2
cups of water.
2(2) + y = 8
4 + y = 8
y = 4
The large pitcher
holds 4 cups of
water.
11. Real life application
• A test has twenty questions worth 100 points. The test consists of
True/False questions worth 3 points each and multiple choice
questions worth 11 points each. How many multiple choice questions
are on the test?
Let x = T/F questions
Let y = Multiple
Choice questions
x + y = 20
3x + 11y = 100
x + y = 20
3x + 11y = 100
3(x + y = 20)
3x + 11 y = 100
3x + 3y = 60
3x + 11y = 100
-8y = -40
8y = 40
y = 5
There are 5 multiple choice questions.
x + 5 = 20
x = 15
There are 15 T/F questions.
12. Benefits from the lesson
•You could determine variables in two equations
in real life
•You are able to solve using graphing, substitution
and elimination
•You are able to estimate values of variables by
solving a system of inequalities
