Transcript Solving Systems of Linear Equations

Solving Systems of Linear Equations
Addition (Elimination) Method
Tutorial 14c
3 Methods to Solve
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There are 3 methods that we can use to
solve systems of linear equations.
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Solve by the Graphing Method
Solve by the Substitution Method
Solve by the Addition (Elimination) Method
Elimination Method:
Using Addition & Subtraction
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In systems of equations where the
coefficient of the x or y terms are
opposites of each other, you can solve
the system by adding the equations
together.
Adding them together causes one of the
variables to be eliminated.
Elimination Method:
Using Addition
Use elimination to solve the system of
equations: x – 3y = 7 and 3x + 3y = 9.
 Check to see if 2 like-terms have opposite
coefficients:
The 1st equation has a –3y and the 2nd has a +3y.
 Therefore, add the two equations.
x – 3y = 7
+ 3x + 3y = 9
4x = 16
x =4
 Substitute 4 for x in
either original equation
and then solve for the
y coordinate.
 Step 4 is to check the
answer. Click here!
x – 3y = 7
4 – 3y = 7
– 3y = 3
y = -1
Elimination Method:
Using Addition
Use elimination to solve the system of
equations: x – 3y = 7 and 3x + 3y = 9.
 The last step is to check your solution by
substituting both values, x = 4 & y = -1,
in both equations.
x – 3y = 7
4 –3(-1) = 7
4 – (-3) = 7
7 =7
3x + 3y = 9
3(4) +3(-1) = 9
12 + (-3) = 9
9 =9
The solution of the system
is (4, -1).
Elimination Method:
Using Multiplication with Addition
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Some systems of equations cannot be solved
by simply adding the equations.
In many cases one or both equations must
first be multiplied by a number before the
system can be solved by elimination.
Consider the following examples.
Elimination Method:
Using Multiplication with Addition
Use elimination to solve the system of
equations: -x + 5y = -2 and 2x + 5y = 4.
 Check to see if 2 like-terms have opposite
coefficients: No terms have opposite coefficients.-1(-x + 5y) = (-2)-1
However, each equation has a 5y!
x – 5y = 2
Therefore, Multiply one of the equations by –1
to get a 5y and a –5y.
 Then add the equations.
-x + 5y = -2
 Substitute 2 for x in
x – 5y = 2
either original equation
-2
+
5y
=
-2
+ 2x + 5y = 4
and then solve for the
3x = 6
y coordinate.
5y = 0
x =2
 Step 4 is to check the
answer. Click here!
y=0
Elimination Method:
Using Multiplication with Addition
Use elimination to solve the system of
equations: -x + 5y = -2 and 2x + 5y = 4.
 The last step is to check your solution by
substituting both values, x = 2 & y = 0,
in both equations.
-x + 5y = -2
-2 +5(0) = -2
-2 + 0 = -2
-2 = -2 
2x + 5y = 4
2(2) +5(0) = 4
4 + 0 =4
4 =4
The solution of the system
is (2, 0).
Elimination Method:
Using Multiplication with Addition
Use elimination to solve the system of
equations: x + 10y = 3 and 4x + 5y = 5.
 Check to see if 2 like-terms have opposite
-4(x
coefficients:
No terms have opposite coefficients -4x
Therefore, multiply the first equation by -4
 Then add the equations.
 Substitute 1/5 for y in
-4x – 40y
+4x + 5y
35y
y
= -12
= 5
= 7
=1/5
+ 10y) = ( 3)-4
– 40y = -12
x + 10y = 3
either original equation
and then solve for the x + 10(1/5) = 3
x coordinate.
x + 2 =3
 Step 4 is to check the
x =1
answer. Click here!
Elimination Method:
Using Multiplication with Addition
Use elimination to solve the system of
equations: x + 10y = 3 and 4x + 5y = 5.
 The last step is to check your solution by
substituting both values, x = 1 & y = 1/5,
in both equations.
x + 10y = 3
1 +10(1/5) = 3
1+ 2 =3
3 =3
4x + 5y = 5
4(1) +5(1/5) = 5
4 + 1 =5
5 =5
The solution of the system
is (1, 1/5)
Elimination Method:
Using Multiplication with Addition
Use elimination to solve the system of equations:
2x – 3y = 8 and 3x – 7y = 7.
 Check to see if 2 like-terms have opposite
-3(2x – 3y) = (8)-3
coefficients:No terms have opposite coefficients. -6x +
Therefore, multiply the first equation by –3 2(
3x –
and the second equation by 2.
9y = -24
7y) = (7)2
6x – 14y = 14
 Then add the equations.
2x – 3y = 8
-6x + 9y = -24  Substitute 2 for y in
2x – 3(2) = 8
either original equation
+ 6x – 14y = 14
and then solve for the
2x – 6 = 8
-5y = -10
x coordinate.
2x = 14
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Step
4
is
to
check
the
y =2
answer. Click here!
x=7
Elimination Method:
Using Multiplication with Addition
Use elimination to solve the system of equations:
2x – 3y = 8 and 3x – 7y = 7.
 The last step is to check your solution by
substituting both values, x = 7 & y = 2,
in both equations.
2x – 3y = 8
2(7) –3(2) = 8
14 – 6 = 8
8 =8
3x – 7y = 7
3(7) –7(2) = 7
21 – 14 = 7
7 =7
The solution of the system
is (7, 2).
Elimination Method:
Using Multiplication with Addition – In Review
Steps for the Multiplication with Addition Method
1. Check to see if any like-terms have opposite
coefficients. If necessary, multiply each side of
either or both equations by numbers that will
make opposite coefficients for one variable.
2. Add to eliminate on of the variables. Solve the
resulting equation.
3. Substitute the known value of one variable in
either of the original equations of the system.
Solve for the other variable.
4. Check the answer by substituting both values in
both equations of the system.