Download Substitution Method

Wednesday: Factoring Warm-up
Find the factors and solutions
x  2 x  35  0
25 x  1  0
3x  x  2  0
9x  6x  0
2
2
2
2
x  2 x  35  0
25 x  1  0
( x  5)( x  7)
(5 x  1)(5 x  1)
{5, 7}
1 1
{ , }
5 5
2
3x  x  2  0
(3x  2)( x  1)
2
2
{ ,1}
3
2
9x  6x  0
3x(3x  2)
2
2
{0,  }
3
Solving Linear Systems
Algebraically
with Substitution and Elimination
Section 3-2
Pages 160-1-67
Objectives
• I can use the Substitution Method to solve
systems of equations
• I can use the Elimination Method to solve
systems of equations
Substitution Method
• Goal
• 1. Isolate one variable in one equation
• 2. Substitute into the other equation(s)
• AWAYS pick the easiest equation to
isolate.
Which Equation to Isolate
2x  3y  9
x  2y  8
x  2y 8
Which Equation to Isolate
y  6x  9
2x  4 y  8
y  6x  9
Which Equation to Isolate
 5 x  y  12
 2 x  4 y  10
y  5 x  12
Which Equation to Isolate
4x  y  3
x  4y  8
y  4 x  3
x  4y 8
Example 1
2x  5 y  7
2(4 y  2)  5 y  7
8 y  4  5 y  7
3 y  3
y  1
x  4y  2
x  4 y  2
x  4(1)  2
x6
(6, 1)
What does it mean?
• When we found the solution (6, -1)
• What does that really mean???
• Intersection of the 2 graphs!!
2
7
2x  5 y  7  y   x 
5
5
1
1
x  4y  2  y   x 
4
2
yaxis
1
0 -9 -8 -7 -6 -5 -4 -3 -2 -1
y=-1/4x+1/2
4
3
2
1
0
y=-2/5x+7/5
9
8
7
6
5
0 -1 1 2 3 4 5 6 7 8 9 1
0
-2
-3
-4
-5
(6, -1)
-6
-7
-8
-9
xaxis
Your Turn!
Solve by Substitution
#1 Homework
2x  5 y  7
x  4y  2
Example 2
2x  5 y  7
2(4 y  2)  5 y  7
8 y  4  5 y  7
3 y  3
y  1
x  4y  2
x  4 y  2
x  4(1)  2
x6
 6, 1
Elimination Method
• GOAL
• 1. Add the equations together and have one
variable term go away.
• 2. Sometimes you will have to multiply
one or both equations by a number to make
this happen.
Multiplying by a number?
• Many times you cannot add the
equations and have a variable term
cancel
• For these cases, you must multiply
One or Both equations by a
number first
• Let’s look at a couple
What to Multiply by?
x-variable will cancel
y-variable will cancel
2x  3y  9
5x  y  4
 5(2 x  3 y  9)
2(5 x  y  4)
(2 x  3 y  9)
3(5 x  y  4)
Example 1
3 x  5 y  4
2 x  3 y  29
2(3x  5 y  4)
3(2 x  3 y  29)
2 x  3(5)  29
6 x  10 y  8
6 x  9 y  87
2x 15  29
2x  14
x7
19 y  95
(7, 5)
y  5
Your Turn
#5 Homework
• Solve the following system of equations
using elimination:
1
2 x  6 y  17
Solution : ( , 3)
2
2 x  4 y  13
Other Methods
• Remember, the solution to a system of
equations if an ordered pair
• You know 2 other methods to check your
answers:
– Graphing Calculator and asking for the
intersection (2nd, Trace, Intersection, E, E, E)
– Substitution Method
Solution Types
Remember there are 3 types of solutions
possible from a system of equations!
No Solution vs Infinite
• How will you know if
you have No Solution or
Infinite Solutions when
solving by Substitution??
Remember Back to Solving
Equations
No Solution
Infinite Solutions
• Variables are gone and
you get this:
• Variables are gone and
you get this:
• 2x + 3 = 2x – 4
• 3 = -4
• This is not possible, so
• 2x + 3 = 2x + 3
• 3=3
• This is always true, so
• No Solution
• Infinite Solutions
Homework
• WS 6-2
• Quiz Next Class