6.3 solving systems using elimination: solution of a system of linear equations: any ordered pair...
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6.3 Solving Systems Using Elimination:
Solution of a System of Linear Equations:Any ordered pair that makes all the equations in a system true.
Substitution Method: Isolate a variable in an equation and substitute into the other equation.Elimination Method: Eliminating one variable at a time to find the solution to the system of equations.
Remember:
GOAL:
USING ELIMINATION: To solve a system by the elimination method we must:
1) Pick one of the variables to eliminate
2) Eliminate the variable chosen by converting the same variable in the other equation its opposite(i.e. 3x and -3x)
3) Add the two new equations and find the value of the variable that is left.
USING ELIMINATION: Continue
5) Check, substitute the values found into the equations to see if the values make the equations TRUE.
4) Substitute back into original equation to obtain the value of the second variable.
NOTE:
Ex:to eliminate 5, we add -5 x, we add –x3y, we add -3y -3.5x, we add 3.5x
In order to eliminate a number or a variable we add its opposite.
TRY IT: What do you add to eliminate: a) 30xy b) -1/2x c) 15y
SOLUTION: a) -30xy b) +1/2x c) -15y
Ex: What is the solution of the system? Use elimination.
2 5 17
6 5 9
x y
x y
USING ELIMINATION: we carry this procedure of elimination to solve system of equations.
SOLUTION:1) Pick one of the variable to eliminate. Looking at the system, y will be easy to eliminate. 2 5 17
6 5 9
x y
x y
2) Eliminate the variable chosen by converting the same variable in the other equation its opposite.
In our system this is already done since +5y and -5y are opposites.
SOLUTION:
8 8x 8 0 8x
1x
3) Add the two new equations and find the value of the variable that is left.
2 5 17
6 5 9
x y
x y
Add them: +
SOLUTION:4) Substitute back into original equation to obtain the value of the second variable.
2 5 17
6 5 9
x yand
x y
1x
2( ) 5 11 7y 2 5 17y
3y Solution: (1, 3)
6( ) 5 91 y
3y
OR
5 15y 6 5 9y 5 15y
SOLUTION:5) Check: substitute the variables to see if the equations are TRUE.
1
3
x
y
2( ) 5( 11 3) 7 2 15 17 TRUE
6( ) 5( 91 3) 6 15 9
and
TRUE
2 5 17
6 5 9
x yand
x y
YOU TRY IT: What is the solution of the system? Use Elimination.
2 7
2 1
x y
y x
SOLUTION:1) Pick one of the variable to eliminate. Looking at the system, y will be easy to eliminate. 2 7
2 1
x y
x y
2) Eliminate the variable chosen by converting the same variable in the other equation its opposite.
In our system this is already done since -y and +y are opposites.
SOLUTION:
4 8x 4 0 8x
2x
3) Add the two new equations and find the value of the variable that is left.
2 7
2 1
x y
x y
Add them: +
SOLUTION:4) Substitute back into original equation to obtain the value of the second variable.
2 7
2 1
x yand
y x
2x
2( 72) y 4 7y
3y Solution: (2, 3)
3y
OR
3y
2( ) 12y
4 1y
SOLUTION:5) Check: substitute the variables to see if the equations are TRUE.
2
3
x
y
2( ) (2 ) 73 4 3 7
TRUE
( ) 2( 13 2) 3 4 1
and
TRUE
2 7
2 1
x yand
y x
7 7 3 3
VIDEOS:Elimination
https://www.khanacademy.org/math/algebra/systems-of-eq-and-ineq/fast-systems-of-equations/v/solving-systems-of-equations-by-elimination
CLASSWORK:
Page 378-380
Problems: As many as needed to master the
concept.