DQ 2 First Question:
I thought since you all are finding examples from the book for others to solve
that I would give a similar one too. Try this one if you like for participation.
John is mixing up an acid solution that needs to be 5% acid and 50 gallons in
the end. He has an acid mixture that is 2% acid and one that is 10% acid.
How much of each should he use to get his final desired 50 gallons of 5%
acid?
Hint: John will need more of the 2% solution than the 10% solution because
5% is closer to 2% than 10%. So he needs more of the weaker acid
solution. If he needed a final solution that was 6% acid he could use half 2%
and half 10% because 6% is halfway between 2% and 10%, but he needs a
5% solution in the end.
X+y=50
0.1x+0.02y = 50*0.05
0.1(50-y) + 0.02y = 2.5
5 – 0.1y + 0.02y = 2.5
-0.08y = -2.5
Y = 31.25
X = 50-31.25 = 18.75
18.75 gal of the stronger, 31.25 gal of the weaker
Next question:
The St. Mark’s Community Barbecue served 250 dinners. A child’s plate cost $3.50 and an
adult’s plate cost $7.00. A total of $1437.50 was collected. How many of each type of plate was
served?
A+c=250 (total dinners)
3.5c + 7a = 1347.5 (price)
Multiply the first by 3.5:
3.5c+3.5a = 875
Subtract that from the second:
3.5a = 472.5
Divide by 3.5:
A = 135
Use the equation for c:
135+c=250
C = 115
115 children, 135 adults
Next question:
Blending Granola. Deep Thought Granola is 25% nuts and dried fruit. Oat
Dream Granola is 10% nuts and dried fruit. How much of Deep Thought and
how much of Oat Dream should be mixed to form a 20-lb batch of granola that
is 19% nuts and dried fruit?
0.25D + 0.1O = 20(0.19) -- based on the nuts and fruit
D + O = 20 --- based on the total weight
Subtract O:
D = 20-O
Plug into the first one:
0.25(20-O) + 0.1O = 3.8
5-0.25O + 0.1O = 3.8
-0.15O = -1.2
O=8
So, D = 20-O = 20-8 = 12
Oat Dream = 8
Deep Thought = 12
Next question:
Here is an example to try. It is example 1 on page 573:
Butterfly Exhibit. White River Gardens in Indianapolis, Indiana,
presents a yearly butterfly exhibit. Pupae from Kenya, Tanzania, Costa
Rica, and Florida are delivered semiweekly. They are stored on rods in a glass
case inside a greenhouse until the metamorphosis takes place. As part of a
recent order, White River Gardens received 63 pupae—morpho granadensis
at $4.15 per pupa and battus polydamus at $1.50 per pupa. The total cost of
the two species was $147.50. How many pupae of each species did the exhibit
receive?
X+y=63
4.15x+1.5y = 147.5
4.15(63-y) + 1.5y = 147.5
-4.15y + 261.45 + 1.5y = 147.5
-2.65y = -113.95
Y = 43
X = 63-43 = 20
20 of the expensive, 40 of the cheap
Next question:
We are asked to figure out two numbers that are placed in a sealed envelope. “Twice the smaller
number is 5 more than two-thirds of the larger number. Three times the larger number is 4 less than
fifteen times the smaller.” What are the numbers?
2x = 5 + 2/3y
3y = 15x – 4
Divide by 3:
Y = 5x – 4/3
Plug that into the first one:
2x = 5 + 2/3(5x-4/3)
2x = 5 + 10/3x – 8/9
-4/3 x = 37/9
X = -37/12
Y = 5(-37/12) – 4/3
Y = -67/4
The numbers are -37/12 and -67/4
Next question:
Mixing Cleaning Solutions.
King’s Service Station uses two kinds of cleaning solution containing acid and
water.“Attack” is 2% acid and “Blast”is 6% acid. They want to mix
the two to get 60 qt of a solution that is 5% acid. How many quarts of each should they
use?
x+y = 60
0.02x + 0.06y = 60*0.05
substitution:
x = 60-y
0.02(60-y) + 0.06y = 3
1.2 - 0.02y + 0.06y = 3
0.04y = 1.8
y = 45
x = 60-y = 60-45 = 15
So:
15 qt of attack
45 qt of blast
Next question:
Find a problem in the text that is similar to examples 2, 3, and 4. Post the problem for
your classmates to solve.
Example-5 and equations 1 and 2 are similar to examples 2,3, and 4 along with the 2
equations each. This one is asking to familiarize and translate as well. If you read below
you can see what they are asking for.
1. A brother of someone drives a car 55mph and he forgets his suitcase. His brother
doesn't discover this until 1 hour after. He drives at 65mph to catch up. D= distance and
t= time in hours.
2. To translate we must know the formula to get is D= RxT (distance equals rate
multiplied by time). We have to get his formula down for the brother and the other
brother who is catching up to him which would be you. So basically just type down the
formula to set up how LONG it would take to catch up to the brother's MILES.
Set up an equation, using t = 0 as the time that you leave:
55(t+1) = 65t
Distribute:
55t + 55 = 65t
Subtract 55t:
10t = 55
Divide by 10:
t = 5.5
It will take 5.5 hours to catch up.
Next question:
Mountainside Fleece sold
40 neckwarmers. Solid-color neckwarmers sold for
$9.90 each and print ones sold for $12.75 each. In all,
$421.65 was taken in for the neckwarmers. How many
of each type were sold?
X+y=40
9.9x+12.75y=421.65
X=40-y
9.9(40-y) + 12.75y = 421.65
396 – 9.9y + 12.75y = 421.65
25.65 = 2.85y
Y=9
X = 40-9 = 31
X = 31 (solid), y = 9 (print)
Solving by graphing
DQ 1
Solving by substitution
Solving by elimination
Which method do you think would be best to solve this system?
y=10x-8
y=1/2x+5
This would be good for substitution, since the “y” is isolated in both equations. You can just set the right
sides of each equal to each other.
The next question :
When you solve a linear system of equations with 2 variables there are three
possible outcomes: 1) There is one solution (the lines intersect at one point);
2) there is no solution (the lines are parallel); 3) There are infinitely many
solutions (the lines are actually the same line just written differently.
Can you tell which is which from the 3 systems of equations below?
A) 3x-9y=10
y=1/3x+5
No solution, because if you multiply the second equation by 9, it’s inconsistent
with the first one
B) 2x+4y=8
x+2y=4
Infinitely many, because the first equation is twice the second.
C) y=2x-7
4x-5y=10
One unique solution
The next question:
Here is a little info on which is the best method for solving systems of
equations. It might be helpful for use on DQ 1 response this week, also.
You look to see which would be the easiest best by how the equations are set
up.
If the two equations are already in y=mx+b format, graphing would be easy
because then the slope is m and the y-intercept is b and those are already
showing from the set up of the equations. As in
y=2x+7
y=-8x-5
I know that I could graph the first one by starting on the y-axis at 7 and then
going up 2 units and to the right 1 unit to get to another point on the line since
2 is the slope and 2=2/1.
Graph the other one in a similar way and see where they intersect. That is the
solution.
When you have something like
2x-8y=15
3x+8y=20
it is totally set up for elimination because you can simply add the 2 equations
to eliminate the y variables since the coefficients on the y are -8 and 8 which
are opposites.
Add the equations:
5x = 35
X=7
Get y:
2(7) – 8y = 15
14 – 8y = 15
-8y = 1
Y = -1/8
X = 7, y = -1/8
When you have something like
x=2y-5
x+y=10
this is set up for substitution since the first equation is solved for x already.
Plug the first into the second:
2y-5+y=10
3y-5 = 10
3y = 15
Y=5
x = 2(5)-5 = 10-5 = 5
x = 5, y = 5
Feel free anyone to solve any of these for participation credit. Show a check
of your solution.
The next question:
Here are some systems of equations. For each system, tell which method
you think is the best to solve it with and then solve it with that method.
2x-4y=7
3x+4y=20
Elimination:
Add the equations:
5x = 27
X = 27/5
Get y:
2(27/5) – 4y = 7
54/5 – 4y = 7
-4y = -19/5
Y = 19/20
So:
X = 27/5, y = 19/20
y=8x+7
y=8x-5
You can see that the slopes are the same, so the lines are parallel.
The y intercepts are different, so they won’t intersect.
NO solution
x=10-y
2x+3y=15
Substitution:
2(10-y)+3y = 15
20 – 2y + 3y = 15
Y = -5
X = 10-(-5) = 15
X = 15, y = -5