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Warm-Up
• Tickets for a concert cost $10 each if you order them online, but
you must pay an $8 service charge per order. The tickets are $12
each if you buy them at the door on the night of the concert.
•
• Write a system of equations to model the situation. Let y be the
total cost. Let x be the number of tickets purchased.
y  10 x  8
y  12 x
6.2: Solving Systems
Algebraically
Substitution method
You can solve linear systems by solving one of the equations for one of
VARIABLES
the ______________.
SUBSTITUTE the expression for that variable into the other
Then, ______________
SUBSTITUTION METHOD
equation. This is called the ________________________.
SOLVED ,
When a system has at least one equation that can be easily __________
for a variable, the system can be solved efficiently using substitution.
 y  5x
Example: What is the solution to the system 
 x  y  12
STEP 1: Substitute 5x for y
in the second equation.
STEP 2: Solve the
equation for x.
STEP 3: Replace x and
solve for y in the first
equation.
STEP 4: Solution to the system:
x  (5 x)  12
6 x  12
x2
y  5(2)
y  10
x=2
y = 10
Solving for a Variable and Using Substitution:
7 x  3 y  2
Example: What is the solution to the system 
2 y  x  6
2 y  x  6
x  2y  6
STEP 1: Rewrite one of the
equations so a variable is by itself.
STEP 2: Substitute this value into
the other equation.
7(2 y  6)  3 y  2
14 y  42  3 y  2
11 y  44
y4
STEP 3: Solve for the variable.
STEP 4: Replace in the first equation.
STEP 5: Solution to the system:
x=2
y =4
7 x  3(4)  2
x2
Special Systems
IDENTITY
If you get an
like 5 = 5 when solving a system this
means there are an infinite number of solutions to the system.
If you get a FALSE statement like 0 = -2 then there are no
solutions to the system.
Examples
1) How many solutions does each system have?
 y  4x  9
 x  3 y  4
b.)
a.) 
6 y  2 x  8
 y  4x  6
6 y  2(3 y  4)  8
6y  6y 8  8
88
Infinitely Many
Solutions
(4 x  9)  4 x  6
4x  9  4x  6
9  6
No Solution
Homework: Sect. 6.2 p. 393-395 #’s 12-24 even