Download 2(3x+7y=188)

The Problem
Stephanie is trying to buy oboe reeds. She
can play using either a medium reed, or
medium-hard reed. At the online store, Reedbay, she can get 4 medium reeds and 2
medium-hard reeds for $109.50. On the other
hand, she could get 3 medium reeds and 7
medium-hard reeds for $188.00. How much
does each reed type cost?
Substitution
Substitution tends to be tricky and complicated. It has several steps, making
errors almost inevitable. Choosing the wrong variable or problem can make
substitution difficult and confusing, proving that elimination is the better
method.
4x+2y=109.5
3x+7y=188
First, you have
to choose one
of the two
equations and
isolate a
variable. In this
case, x is used.
4x+2y=109.5
-2y
-2y
4x=109.5-2y
/4
/4
x=109.5-2y
4
3(109.5-2y)+7y=188
4
82.1-1.5y+7y=188
82.1+5.5y=188
-82.1
-82.1
5.5y=105.9
/5.5
/5.5
y=19.25
Next you plug the
resulting solution into the
other equation and solve
for the other variable, y.
4x+2(19.25)=109.5
4x+38.5=109.5
-38.5 -38.5
4x=71
/4 /4
x=17.75
(17.75, 19.25)
You then have to plug in the
solution you got for the
latter variable, y, into the
first equation and solve the
equation for the original
variable, x.
Elimination
As you can see,
elimination is
clearly the better
method. It has a
simple concept and
has fewer steps
than substitution
and graphing. This
can help prevent
any mistakes that
might be made.
The first step is to combine the two
equations. To do this you multiply the
whole problems, so that when they’re
4x+2y=109.5
added together at least one variable will be 3x+7y=188
canceled. From there, you simplify the
resulting equation until you find a solution 7(4x+2y=109.5)
-2(3x+7y=188)
for the remaining variable, x.
28x+14y=766.5
-6x+(-14y)=-376
22x=390.5
/22 /22
x=17.75
Then you plug the variable’s
solution into one of the
problems and solve for the
other variable, y.
3(17.75)+7y=188
53.25+7y=188
-53.25 -53.25
7y=134.75
/7
/7
y=19.25
(17.75, 19.25)
Graphing
Prices can be particularly hard to deal
with when using graphing because
they don’t always divide evenly.
Unlike elimination, graphing requires
the two equations to be in slope
intercept form, or, y=mx+b.
4x+2y=109.5
3x+7y=188
4x+2y=109.5
-4x -4x
2y=-4x+109.5
/2
/2
y=-2x+54.75
3x+7y=188
-3x
-3x
7y=-3x+188
/7
/7
y=-3/7x+188/7
Elimination...in Oboe & in all Oboe-Related
Activities… Inc. is a corporation that deals
with math in oboe related situations. This
includes reeds, cases, cleaning cloths, cork
grease, actual oboes, sheet music, and,
among other things, oboe ninjas. They’re
everywhere. And if you don’t use
elimination, they WILL be angered.
Depression hurts. Not using
elimination could hurt even
more.
Elimination…In Oboe & in all Oboe-Related Activities…Inc.
But what if you had a different problem?
What if your equations were:7x + 2y = 16
Elimination
–21x – 6y = 24
3(7x + 2y = 16)
–21x – 6y = 24
21x+6y=48
-21x-6y=24
0=72
This equation has no solution. The lines
would be parallel.
Substitution
7x + 2y = 16
–21x – 6y = 24
7x+2y=16
-7x
-7x
2y=-7x+16
/2
/2
y=-(7/2)x+8
-21x-6(-(7/2)x+8)=24
-21x+21x-48=24
-48=24
This nonsense answer means that,
again, there is no solution and that
the lines would be parallel.
Graphing
7x + 2y = 16
–21x – 6y = 24
7x+2y=16
-7x
-7x
2y=-7x+16
/2
/2
y=-3.5x+8
-21x-6y=24
+21x
+21x
-6y=21x+24
/-6
/-6
y=-3.5x-3
As you can see, the lines are
parallel. This means that the
equation has no solution.
Or maybe the equations are: 12x+5y=24
Elimination
24x+10y=48
-2(12x+5y=24)
24x+10y=48
-24x-10y=-48
24x+10y=48
0=0
This equation has infinitely many solutions.
Both equations represent the same line. This
can be written as (x,y) 12x+5y=24.
Graphing
12x+5y=24
24x+10y=48
12x+5y=24
-12x
-12x
5y=-12x+24
/5
/5
y=-2.4x+4.8
24x+10y=48
-24x
-24x
10y=-24x+48
/10
/10
y=-2.4+4.8
As you can see, these equations represent the same line.
Therefore there are infinitely many solutions.
Substitution
12x+5y=24
24x+10y=48
12x+5y=24
-5y
-5y
12x=24-5y
/12
/12
x=2-(5/12)y
24(2-(5/12)y)+10y=48
48-10y+10y=48
48=48
-48 -48
0=0
This equation has infinitely many
solutions, meaning that they each
represent the same line.
In conclusion…
Elimination is the best method for
working these problems. It has a
simple concept and uses a minimal
amount of work. Since there are
fewer steps than substitution and
graphing, there is not quite as much
room for error.