Math word problem plz help me out

Question

A jewelry box contains 6 gemstones: moldavites, amber, and amethysts. The moldavite stones weigh 2 grams and cost 25 dollars, the amber stones weigh 5 grams and cost 20 dollars and the amethyst stones weigh 200 grams and cost 15 dollars. In total, all the gemstones weigh 222 grams and cost 120 dollars. Determine how many of each type of gemstone is in the box. 

Answer 1

Let x = the number of moldavites, y = the number of amber and z = the number of amethysts.

From the description

x+y+z=6

2x+5y+200z=222

25x+20y+15x=120

[ 1 1 1 6]

[ 2 5 200 222]

[25 20 15 120]

R2 = R2-2R1, R3=R3-25R1

[ 1 1 1 6]

[ 0 3 198 210]

[0 -5 -10 -30]

R2 = R2/3, R3 = R3/-5

[1 1 1 6]

[0 1 66 70]

[0 1 2 6]

R3 = R3-R2

[1 1 1 6]

[0 1 66 70]

[0 0 -64 -64]

Back solving

z = -64/-64 = 1

y+66 = 70 --> y = 4

x + 4 + 1 = 6 -->x = 1

There is one moldavite, four amber and one amethyst.

Answer 2

x=moldavits

y=amber

z=amethyst

x+y+z = 6

2x+5y+200z = 222

25x+20y+15z=120 --> 5x+4y+3z=24

x + y + z = 6

2x + 5y + 200z = 222

5x + 4y + 3z = 24

-2 * row1 + row2;-5*row1+row3

-------------------------------

3y + 198z = 210

-y - 2z = -6 -->y+2z= 6

y=6-2z

3(6-2z)+198z =210

18 -6z+198z=210

18 +192z =210

192z= 192

z=1

then y =6*2*1=4

and x+4+1=6 --> x=1

x=1, y=4,z=1

Answer 3

Hi Waker!

Let's try to express the information we have in terms of equations.

Let's call m the number of moldavites, a the number of ambers, and t the number of amethysts. This becomes a set of 3 equations (number of stones, price, weight) and 3 unknowns.

1. Number: we know there are 6 stones total. So m + a + t = 6 [1]
2. Weight: 2 m + 5 a + 200 t = 222 [2]
3. Price: 25 m + 20 a + 15 t = 120 [3]

Now let's try to combine those equations to simplify things a bit. There are many ways to do this. But for instance, let's subtract 2 times [1] from [2] so that the term in "m" disappears:

(2 - 2 * 1) m + (5 - 2 * 1) a + (200 - 2 * 1) t = 222 - 2 * 6

which leads to

3 a + 198 t = 210 [4]

We also know that a, m and t are integers (we only have whole stones), and we know none of the three is larger than 6 (we only have 6 stones), so [4] tells us what a and t are. t can only be 1. If t were zero, then we would have 3 a = 210, which is impossible since a can't be more than 6. If t were at least 2, then we would have 3 a + 396 = 210, and a would need to be negative, which is not possible.

So, t has to be 1, which means (from [4]) that a must be (210 - 198) / 3 = 4

Then, from [1], we get that m is 1 as well.

In short, there are 1 moldavite, 4 ambers, and 1 amethyst.