7-2 and 7-3 solving systems of equations algebraically the substitution and elimination method

7-2 and 7-3Solving Systems of

Equations Algebraically

The Substitution and Elimination Method

Solving Systems of Equations

• You have already seen how to solve a system of equations by graphing but you won’t necessarily always have a graphing calculator. So there are 2 other ways to solve systems by using algebra…

• The 2 other ways to solve systems of equations:– Substitution– Elimination

Solving Systems of Equations

• Substitution is used when the equation is already solved for one variable then plug-n-chug that value into the other equation.

• Elimination is when you solve for one variable by adding, subtracting, or multiplying to the original equations.

SubstitutionEx. 1 Consider the system

x - 2y = 5

y = 2x + 3

REMEMBER: We are trying to find the Point of Intersection. (x, y)

Substitution is desired if one variableis already by itself or can be solvedfor very easily. What is already by itselfon one side of the equation?

So take y = 2x + 3 and plug it in for y in the other equation.

x – 2y = 5x – 2(2x+3) = 5x – 4x – 6 = 5-3x – 6 = 5 x = -3 or -11/3

SubstitutionConsider the system

x - 2y = 5

y = 2x + 3


Now that you’ve found the answer forx = -3 or -11/3 , how do you think you’ll find the answerfor y?

So what you got for x and plug it in and find y. You’ll now have your point of intersection.

y = 2x + 3y = 2(-11/3) + 3y = -4 or -13/3So your answer is(-11/3, -13/3)


y = x + 1

y = 2x - 1


Substitution is desired if one variableis already by itself or can be solvedfor very easily. What is already by itselfon one side of the equation?

So take both equations and set them equal to each other since both equal y.

x + 1 = 2x – 1-x -x 1 = x – 1 +1 +1 x = 2

SubstitutionConsider the system

y = x + 1

y = 2x - 1


Now that you’ve found the answer forx = 2, how do you think you’ll find the answer for y?

So what you got for x and plug it in and find y. You’ll now have your point of intersection.

y = x + 1y = 2 + 1y = 3Your answer is(2,3)


x - y = 1

x = ½ y + 2


Now you try this one all on your own.

What is your final answer? (3,2)

Now we’re going to look at the Elimination

Method

Elimination using Addition

Ex. 4 Consider the system

x - 2y = 5

2x + 2y = 7


Lets add both equations to each other


Consider the system…what makes it different than the othersystems we just solved for?

x - 2y = 5

2x + 2y = 7

Lets add both equations to each other+

NOTE: We use the Elimination Method, if we can immediately cancel out two like terms.


Consider the system

x - 2y = 5

2x + 2y = 7

Lets add both equations to each other+

3x = 12x = 4

ANS: (4, y)



Consider the system

x - 2y = 5

2x + 2y = 7

ANS: (4, y)

Lets substitute x = 4 into this equation.

4 - 2y = 5 Solve for y - 2y = 1

y = 12



Consider the system

x - 2y = 5

2x + 2y = 7

ANS: (4, )

Lets substitute x = 4 into this equation.

4 - 2y = 5 Solve for y - 2y = 1

y = 12

12




3x + y = 14

4x - y = 7



Consider the system

3x + y = 14

4x - y = 7

7x = 21x = 3

ANS: (3, y)

+


Consider the system

ANS: (3, )

3x + y = 14

4x - y = 7

Substitute x = 3 into this equation

3(3) + y = 149 + y = 14

y = 5

5


Elimination using Multiplication


6x + 11y = -5

6x + 9y = -3+12x + 20y = -8 When we add equations together,

nothing cancels out


Consider the system

6x + 11y = -5

6x + 9y = -3


Consider the system

6x + 11y = -5

6x + 9y = -3

-1 ( )


Consider the system

- 6x - 11y = 5

6x + 9y = -3+-2y = 2

y = -1

ANS: (x, )-1


Consider the system

6x + 11y = -5

6x + 9y = -3

ANS: (x, )-1

y = -1

Lets substitute y = -1 into this equation

6x + 9(-1) = -36x + -9 = -3

+9 +9

6x = 6x = 1


Consider the system

6x + 11y = -5

6x + 9y = -3

ANS: ( , )-1

y = -1

Lets substitute y = -1 into this equation

6x + 9(-1) = -36x + -9 = -3

+9 +9

6x = 6x = 1

1



x + 2y = 6

3x + 3y = -6

Multiply by -3 to eliminate the x term


Consider the system

x + 2y = 6

3x + 3y = -6

-3 ( )


Consider the system

-3x + -6y = -18

3x + 3y = -6+-3y = -24

y = 8

ANS: (x, 8)


Consider the system

x + 2y = 6

3x + 3y = -6

ANS: (x, 8)

Substitute y = 8 into equation

y =8

x + 2(8) = 6x + 16 = 6

x = -10


Consider the system

x + 2y = 6

3x + 3y = -6

ANS: ( , 8)

Substitute y = 8 into equation

y =8

x + 2(8) = 6x + 16 = 6

x = -10

-10


x + 2y = 5

2x + 6y = 12

ANS: (3,1 )


HomeworkYou have a worksheet for

classwork/homework TOMORROW. The worksheet does not tell you which method to use. You must figure out if substitution or elimination is the way you want to solve the systems. Whatever you choose is not wrong. It may take longer but

whatever you choose is not wrong! My advice is to go ahead and label S or E on the problems

tonight so you have a plan on how to solve them. Make you life easier for tomorrow!!!

7-2 and 7-3 solving systems of equations algebraically the substitution and elimination method

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