URGENT... Please help solve.... I AM GOING CRAZY!!!!?

Question

1. How many solutions does the following system have:
6x - 3y = 9
2x - y = 3
Explain how to determine the number of solutions without solving the system. Then apply elimination, and interpret the resulting expression.

2. How many solutions does the following system have:
4x + 8y = 16
3x + 6y = 9
Explain how to determine the number of solutions without solving the system. Then apply elimination, and interpret the resulting expression.

3.
Translate the following into a system of equations, then solve it:
A customer walks into an electronics store and buys five MP3 players and eight sets of headphones, paying $840. A second customer buys three MP3 players and four sets of headphones, and pays $480. How much does an MP3 player cost? How much does a set of headphones cost?

Answer 1

Reviewing the material you were supposed to have learned before doing this problem is a more useful approach than going crazy or even asking here.

Answer 2

1. 6x - 3y= 9 -----> 2x - y = 3 (Divide all by 3)
This is the same as the second equation 2x - y = 3

Thus there are an infinite set of solutions where y = 2x - 3 in both cases These types of equations are called indeterminate as one is a multiple of the other.

Graphically they would produce the same line. Therefore there really is only one equation there (the other being a variation of the original). Hence ALL points on the line satisfy both conditions

2. 4x + 8y = 16 ----> x + 2y = 4 (Divide all by 4)
3x + 6y = 9 ----> x + 2y = 3 (Divide all by 3)
There is an inconsistency here as x + 2y has to equal both 3 and 4 and this is impossible. These equations are callled inconsistent.

Graphically these would be a pair of parallel lines that never intersect and therefore there is no point on the number plane that can satisfy BOTH conditions simultaneously.

3 Let the cost of an MP3 player be $M and the cost of a pair of headphones be $H

Then the 1st condition says

5M + 8H = 840 ...... (1)

and the second conditon says

3M + 4H = 480 ...... (2)

Note that in (1) there are twice as many headphones as in (2)

So 2 X (2)

6M + 8H = 960 ...... (3)

(well if the second customer bought twice as many MP3 players and headphones (s)he would have paid twice as much!!

Now subtract (1) from (3)

M = 120

Substitute this into (1)

5 X 120 + 8H = 840

ie 600 + 8H = 840

8H = 240

H = 30

Check in 2

3M + 4H = 3 X 120 + 4 X 30 = 360 + 120 = 480 (YESSSSSSSSSSSSS!!!!!!)

Thus the headphones cost $30 (AND the MP3 player cost $120)

Answer 3

For numbers 1 and 2, here's how to do it:

Let's say you are given 2 equations:
ax + by = c
dx + ey = f
where a,b,c,d,e and f are constants.

To tell you, there are only 3 possible number of solutions for this: one (the lines intersect at one point), zero (the 2 lines are parallel) or infinitely many (the 2 lines coincide).

Now here are the conditions:
a) If a/d ≠ b/e, then there is one unique solution.
b) If a/d = b/e ≠ c/f, then there are no solutions.
c) If a/d = b/e = c/f, then there are infinitely many solutions.

Now,
-----------------------------------
1)
6x - 3y = 9
2x - y = 3

The first step is to find a,b,c,d,e and f.
ax + by = c
dx + ey = f

Now,
a = 6
b = -3
c = 9
d = 2
e = -1
f = 3

Now,
a/d = 6/2 = 3
b/e = -3/-1 = 3
c/f = 9/3 = 3

Thus, a/d = b/e = c/f
Therefore, there are infinitely many solutions


---------------------------------------
2)
4x + 8y = 16
3x + 6y = 9

Thus,
a = 4
b = 8
c = 16
d = 3
e = 6
f = 9

Then,
a/d = 4/3
b/e = 8/6 = 4/3
c/f = 16/9

Thus, a/d = b/e ≠ c/f.
Therefore, there is no solution to the system of equation

------------------------------------
3)

Here's how we do it. We assign a variable to represent the unknown price of each item.:

a) If we say that one MP3 player costs p dollars, therefore 5 MP3 players cost 5·p dollars, and 3 MP3 players cost 3·p dollars.

b) If we say that one set of headphones costs h dollars, therefore 8 sets of headphones cost 8·h dollars, and 4 sets of headphones cost 4·h dollars.

c) Now since 5 MP3 players plus 8 sets of headphones cost $840, we can set up the equation:
5·p + 8·h = 840

d) Now since 3 MP3 players plus 4 sets of headphones cost $480, we can set up the equation:
3·p + 4·h = 480

e) Thus, we have the equations:
5p + 8h = 840
3p + 4h = 480

Now we can solve for the value of p and h. But first, from the method used in 1 and 2, we can see that a/d ≠ b/e and therefore there is one unique solution for p and h.
5p + 8h = 840
3p + 4h = 480

Now, if we multiply -2 to both sides of the "2nd" equation,
5p + 8h = 840
-6p - 8h = -960

and add the equations together,
-p = -120

Thus,
p = 120

We solve for h,
3p + 4h = 480
3(120) + 4h = 480
360 + 4h = 480
4h = 480 - 360
4h = 120
h = 120/4
h = 30

Thus, we see that p (the cost of 1 MP3 player) is $120, and h (the cost of 1 set of headphones) is $30.

^_^

Answer 4

I dont know about questions 1 and 2 but I think I worked out question 3 I cant put it into a system of equations but here goes:

An MP3 player costs $120
a set of headphones costs $30

so

first customer $129 x 5 = $600
$30 x 8 = $240

Total $840

second customer $120 x 3 = $360
$30 x 4 = $120

Hope this has helped.

Answer 5

question 3.
5M +8H = 840
3M +4H = 480 multiply by 2 gives 6M+8H = 960
subtract one equation from the other
6M+8H-(5M+8H) = 960-840
M = 120
put that in the first equation
5x120 + 8H = 840
reshuffle
8H = 840 - 600 = 240
so H = 30
check in second equation
3x120 + 4x30 = 360+120=480

question 1.
I don't get it the 2 equations are the same so probably an infinite number of solutions

question 2.
One solution. The is a first order equation (ie no squares or cides) so it will have one solution.

Answer 6

I can't help you with the first part of your question but I can help with the word problem.

First customer buys 5 mp3 players (use 5x) and 8 headphones (use 8y) and he spends 840.00. The equation is:

5x+8y=840

Second customer buys 3 mp3 players (use 3x) and 4 headphones (use 4y) and spends 480.00. The equation is:

3x+4y=480.00

Put the two equations together and we want to cancel out the x's to find the value of y, the headphones. We have to find the least common denominator of the x's. LCM=15. Multiply the first equation by 3 and the second eqation by -5.

(3) 5x+8y=840
(-5) 3x+4y=480

15x+24y=2520
-15x-20y=-2400=

4y=120 Divide both sides by 4
y=30.00

One set of head phones costs $30.00.

Answer 7

1.since there r two variables [x&y]and the degree of the equation is 1{highest power of any variable in an equation is degree of the equation} u can get only one value of x and one of y
6x-3y=9 & y=2x-3 in this case both the eqautions r same so it cant be solved
to solve such equations u need two different equations in x&y with different constants in front of x&y
2.same here , both equations have same coefficients of x &y
3.5x+8y=840 & 3x+4y=480 [x =cost of one mp3 ,y =cost of oneheadphone]
one mp3 costs 120$
one headphone costs 30$

Answer 8

Looks like someone already answered you when you asked this 2 days ago. You should review the answers to your previous same question (see link below).

Answer 9

i am going crazy to ., i saw this question beforwe