Transcript Lesson 4-2: Solving Systems – Substitution & Linear Combinations

Lesson 4-2: Solving Systems –
Substitution & Linear
Combinations
Objectives:
Students will:
Solve systems of equations using
substitution and linear combinations
Substitution (means replace)
1. Solve one equation for one of the variables (find an easy one!)
2. Substitute what that variable equals into the other equation.
3. Solve for the remaining variable.
4. Substitute the value of this variable back into step 1
5. Solve for the other variable.
6. Check that the two values solves both equations
Example 1 Solve the system
5x + 3y = 6
y = -1 - x
5x + 3 (-1 - x) = 6
5x -3 - 3x = 6
2x – 3 = 6
x = 9 or 4.5
2
Wow step 1 is
done!
y = -1 – x
y = -1 - 9
2
y=
11
or  5.5
2
The solution is (4.5, -5.5)
Example 2:
x + 2y = 6
x = 6 - 2y
4x + 3y = 4
4(6x - 2y)+ 3y = 4
This one was easy to solve for
x.
Look for variable with no
coefficient
x = 6 - 2y
24 - 8y + 3y = 4
x = 6 -2(4)
-5y = -20
x = -2
y=4
The solution is (-2, 4)
Example 3
5x + 3y = 17
-5x + 2y = 3
Solving for either variable is a challenge
here!!!
There has to be an easier way!
Linear Combination
(ELIMINATION)
1. Set up both equations in Standard Form : Ax + By = C
2. Obtain opposite x or y terms. Multiply (one or both) equations
by a number if you have to make opposites.
3. Add the equations vertically – x’s or y’s will eliminate
4. Solve for the remaining variable.
5. Substitute the value of this variable back into one of the original
equations
6. Solve for the other variable.
7. Check that the two values solves both equations
Example 3 Use elimination
This is an easy
one x’s are
already opposite
5x +3(4) =17
5x +12 =17
x=1
5x + 3y = 17
+ -5x + 2y = 3
5y = 20
y= 4
The solution is (1,4)
This time let’s make the
y’s opposite. We need to
change both equations
Example 4
6x + 2y = -16 ) 5
( -12x - 5y = 31 ) 2
(
Finishing this
one now will
be a piece of
cake!
30x +10y = -80
+ -24x - 10y = 62
Example 4
1
2
(
x  y 1 ) 6
2
3
3
1
( 4 x  3 y  2 ) 12
Now you can do it!
Give it a try!!!
What would make this easier…
I remember!
Get rid of fractions by
multiplying by LCD