SOLVING SYSTEMS OF EQUATIONS

PROJECT BY HIND ALAWADISHAMMA HASSAN, MAHRA ABDULLATIF, MARYAM ADEL, ALIA

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

Objectives:

•Solve systems of linear equations by: 1. Graphing

2. Substitution3. Elimination

•Solve systems of inequalities

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 solution1. Independent = 1 solution2. Dependent = infinite solutions• Inconsistent: no solution

How to solve a system of equations?1. Graphing

Y=2x+3Y=11-2x

X Y=2X+3

0 2.0+3=3

1 2.1+3=5

0

1

Y=11-2x11-2.0=11

11-2.1= 9

(0,3) (1,5)(0,11) (1,9)

(0,3) (1,5)(0,11) (1,9)

Approximately: (2,7)

How to solve a system of equations?2. Substitution Y=2x+3Y=11-2x

2x+3=11-2x2x+2x=11-34x=84x(/4)=8(/4)X=2

Y=2.(2)+3Y=4+3Y=7

Y=11-2xY=11-2.(2)Y=11-4Y=7

Y=2x+3Y=11-2x

How to solve a system of equations?3.elimination Y=2x+3- Y=-2x+11__________0=4x-8

=4x-88=4x8(/4)=4x(/4)2=x

Y=2.(2)+3Y=4+3Y=7

Y=11-2xY=11-2.(2)Y=11-4Y=7

How to solve a system of inequalities?

6x-3y<-9-3y<-9-6xY<3+2x4x-3y>=1-3y>=1-4xY>=-0.33+1.33x

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 pitchery = large pitcher 2x + y = 8y - x = 2

2x + y = 8 -x + y = 2subtract: 3x = 6 x = 2The small pitcher holds 2 cups of water.

2(2) + y = 84 + y = 8y = 4The large pitcher holds 4 cups of water.

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 questionsLet y = Multiple Choice questionsx + y = 203x + 11y = 100

x + y = 203x + 11y = 1003(x + y = 20)3x + 11 y = 1003x + 3y = 603x + 11y = 100-8y = -408y = 40y = 5There are 5 multiple choice questions.

x + 5 = 20x = 15There are 15 T/F questions.

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

Resources:• http://www.purplemath.com/modules/syseqgen.htm• http://math.about.com/od/linearequations/tp/Syst_Lin-1.html• http://

www.mathwarehouse.com/algebra/linear_equation/systems-of-equation/system-linear-inequality.php• The algebra textbook

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