solving linear systems algebraically with substitution section 3-2 pages 160-1-67
TRANSCRIPT
Solving Linear Systems Algebraically
with Substitution
Section 3-2
Pages 160-1-67
Objectives
• I can use the substitution method to solve equations
• I can solve word problems using Substitution
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
82
932
yx
yx
842
96
yx
xy
1042
125
yx
yx84
34
yx
yx
2 8x y 6 9y x
5 12y x 4 3
4 8
y x
x y
Example 1
2 5 7 4 2x y x y
4 2x y 2( 4 2) 5 7y y
8 4 5 7y y
3 3y 1y
4( 1) 2x
6x
(6, 1)
What does it mean?
• When we found the solution (6, -1)
• What does that really mean???
• Intersection of the 2 graphs!!
2 72 5 7
5 51 1
4 24 2
x y y x
x y y x
1 2 63 4 5 7 8 9 10
4
3
2
7
56
8
9
x-axis
y-axis
0
1-2-6 -3-4-5-7-8-910
-4
-3
-2
-1
-7
-5
-6
-8
-9
0
-1
y=-2/5x+7/5
y=-1/4x+1/2
(6, -1)
Example 2
3 2 3 3 3x y x y
3 3y x 3 2( 3 3) 3x x
3 6 6 3x x
9 3x 1
3x
13( ) 3
3y
2y 1
,23
Your Turn
• Solve the following system of equations using substitution:
124
623
yx
yx)3,0(:Solution
Other Methods
• Remember, the solution to a system of equations if an Ordered Pair
• You know 2 other methods to check your answers:– Graphing to find the intersection– Graphing Calculator and asking for the
intersection (2nd, Trace, Intersection, E, E, E)
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• Variables are gone and
you get this:
• 2x + 3 = 2x – 4• 3 = -4• This is not possible, so
• No Solution
Infinite Solutions• Variables are gone and
you get this:
• 2x + 3 = 2x + 3• 3 = 3• This is always true, so
• Infinite Solutions
Word Problems
• When solving a word problem, consider these suggestions
• 1. Identify what the variables are in the problem
• 2. Write equations that would represent the word problem, looking for key words
• Sum, difference, twice, product, half, etc…
Example 1
• GEOMETRY: The length of a rectangle is 3 cm more than twice the width. If the perimeter is 84 cm, find the dimensions.
Variables:
Length (L)
Width (W)
Equations:
L = 2W + 3
2L + 2W = 84
Now, solve by substitution
Example 2
• Melissa has 57 coins in dimes and nickels. The total value of the coins is $4.60. How many coins of each kind does she have?
Nickels (N)
Dimes (D)
Equations:
N + D = 57
10D + 5N = 460
Now, solve by substitution
Example 3
• At a recent movie, adult tickets were $4.50 and student tickets were $2.50. During opening night a total of 300 tickets were sold earning $1130. How many of each ticket type were sold?
Adult Ticket (A)
Student Ticket (S)
Equations:
A + S = 300
4.50A + 2.50S = 1130
Now, solve by substitution
Homework
• Substitution Worksheet