Solving Systems Using Elimination

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SOLVING SYSTEMS USING ELIMINATION

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Sometimes the elimination method is easier than the substitution method.
The elimination method is so-called because the original system is replaced (if needed) by an equivalent system,
where "addition" of the two equations eliminates one of the variables.

Here's a key concept behind this method:
If A=B and C=D , then A+C=B+D .

Here's the same concept, arranged vertically:

If	A	=	B
and	C	=	D	,
then	A+C	=	B+D	.

This observation allows you to generate a new equation by adding the left-hand and right-hand sides of two existing equations.
This new equation will be true, whenever the original two equations are true.
The method is illustrated below.

EXAMPLE (a system ideally suited to the elimination method):
Solve the system:

3x	-	2y	=	5
-3x	+	7y	=	2

3x	-	2y	=	5
-3x	+	7y	=	2
Add:		5y	=	7

Notice that there is a 3x in the first equation and a -3x in the second equation.
"Adding" the two equations eliminates the variable x.

y=7 5

Solve for y .

3x-2( 75 )=5
3x=5+ 145
3x=39 5
x=39 15

Substitute this value of y back into either original equation and solve for x .

x=13 5

Report all numbers as fractions in simplest form.

The unique solution is the ordered pair (13 5,7 5) .

Clearly report the unique solution.

EXAMPLE (replacing one equation with an equivalent equation):
Solve the system:

2x	-	5y	=	1
6x	+	7y	=	8

-3(2x -5y)=- 3(1) 6x+7y =8	There is a 2x in the first equation and a 6x in the second equation. Here, 6 is a multiple of 2 . Multiply both sides of the first equation by -3 and replace the first equation with this new one.
-6x+15 y=-3 6x+7y =8	In this equivalent system, it is easy to eliminate x .
22y=5 y=5 22	"Add" the two equations to eliminate x and then solve for y .
2x-5( 522 )=1 x=47 44	Substitute this value of y back into a simplest earlier equation, and solve for x .
The unique solution is the ordered pair (47 44,5 22) .	Clearly report the unique solution.

EXAMPLE (replacing both equations with equivalent equations):
Solve the system:

2x	-	3y	=	5
5x	+	4y	=	7

4(2x- 3y)=4( 5) 3(5x+ 4y)=3( 7) 8x-12y =20 15x+12y =21 Add: 23x=41	Looking at the x terms, 5 is not a multiple of 2 . Looking at the y terms, 4 is not a multiple of 3 . We will choose to eliminate the y terms. The least common multiple of 3 and 4 is 12 . Multiply both sides of the first equation by 4 (which will give a y-coefficient of -12 ). Multiply both sides of the second equation by 3 (which will give a y-coefficient of 12 ).
x=41 23	Solve for x .
2(41 23)-3 y=5 y=-11 23	Substitute this value of x back into a simplest earlier equation, and solve for y .
The unique solution is the ordered pair (41 23,-11 23) .	Clearly report the unique solution.

On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.