Transcript Solving Equations Algebraically

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
2x  3y  9
x  2y  8
y  6x  9
2x  4 y  8
x  2y 8
y  6x  9
 5 x  y  12
 2 x  4 y  10
y  5 x  12
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
Example 2
3x  2 y  3
3x  y  3
3x  2(3 x  3)  3
3x  6x  6  3
9x  3
1
x
3
y  3 x  3
1
y  3( )  3
3
y2
1 
 ,2
3 
Your Turn
• Solve the following system of equations
using substitution:
3x  2 y  6
x  4 y  12
Solution : (0, 3)
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
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
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