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Limitations of Graphing
Cannot read the solution. The lines
cross at a point that is not
readable on the graph.
GraphtheSystemofLinearEquations
x y = 3
3x y = 0
y = x - 3
m = 1 (=
1
1
b = -
y = 3x - 0
m = 3 (=
3
1
b = 0
Line 1
Line 2
Solution?
Substitution
TheMethodofSubstitution
Substitution means
to replace something with
it's equal.
Steps
StepsforusingtheMethodofSubstitution
Substitute
that variable (the one that you solved for in step
- in the 2
nd equation with what it equals
Solve for the remaining variable.
- Plug answer into equation from step #
to find other
variable.
- Check that your ordered pair works in both original
equations.
- Solve ONE of the equations for one of the
variables (whichever equation & variable is
easiest to solve for.)
Example
y = 10
10x + 2y = 10
Equation 1 is already solved for "y"
Plug in -10 wherever there is a "y"
in the OTHER equation
10x + 2(-10) = 10
Now, solve for "x"
10x - 20 = 10
+20 +
10x = 30
10 10
x = 3
Now, plug "3" in for
"x" and solve for "y"
10(3) + 2y = 10
30 + 2y = 10
-30 -
2y = -
2 2
y = -
Check that (3, -10)
works in both
equations.
Last Step
SolveusingSubstitution
Example
5x 3y = 23
2x + y = 7
Which equation is easiest to solve
for "x" or "y"?
(4, -1)
Answer
Example
5x + 2y = 12
6x 2y = 14
(2, 1)
Answer
Which equation is easiest to solve
for "x" or "y"?
Example
y = x + 3
2y + 2x = 4
False - ∅
No Solution
Answer
Application
At Amy’s Amusement Park, tickets sell for $24.50 for
adults and $16.50 for children. On Sunday, the
amusement park made $6405 from selling 330 tickets.
How many of each kind of ticket was sold?
What is known?
- 330 tickets were sold
If x = Adult tickets and y = Children tickets
THEN...
x + y = 330
- Adult ticket prices = $24.50 each (ie: 24.50x),
Children ticket prices = $16.50 (ie: 16.50y)
Total $$ Made = 6405
SO...
24.5x + 16.5y = 6405
Example
-2y from both
x 2y = 3
carry prob over x + 2y = 3
x = 3 + 2y
x + 2y = 3
0 = 6
Not True ∅
If the variables cancel out and
- a TRUE statement is left then the lines are the same
and the answer is:
∞
- a FALSE statement is left then the lines are parallel
and the answer is:
∅
Are
the
x's,
y's
&
='s
lined
up??
NOTE!!
Advanced Example
Whatif novariablecancelsout?
x + y = 3
2x + y = 5
Mult every term by -1 x y = 3
Carry problem over
2x + y = 5
x = 2
(2, )
Plug 2 in for "x"
in either equation
and solve for "y"
2 + y = 3
-2 -
y = 1
1
Advanced Example
Trythisone:
4x + 3y = 2
x + 2y = 3
Carry problem over 4x + 3y = 2
Mult every term by -
4x 8y = 12
-5y = -
( , 2)
y = 2
Plug 2 in for "y"
into ANY equation
and solve for "x"
x + 2(2) = 3
-4 -
x = -
-5 -
Homework
Homework:
p141 (1st Column: 13-22,
2nd Column 14-23, 25,
2nd Column: 44-57,
3rd Column 45-58, 63, 66)
