When doing this problem, let's let X be the amount for one gram of gold, Y be the amount for one gram of silver, and Z be the amount for one gram of copper. So when we look at this .75% gold, so that's .75% for one gram of gold plus .05 Y for one gram of silver plus .20 Z for one gram of copper. But it tells us that that first alloy cost $2195.40 for 100 grams. So what's going to happen is I'm going to take all of these and instead of having them for one gram, I'm going to multiply them all by 100. And that basically moves the decimal place over two places. So now this equation is for 100 grams it's 75 X. For 100 grams, it's five Y. For 100 grams it's 20Z and the price for that 100 grams was $2195.40. Doing the same thing for the next equation for one ounce it was .75, but we're going to have 100 oz, so it's going to be 75. For one ounce it's going to be .125 but for 100 oz it's going to be 12.5. Same idea for the copper and that's going to equal 2225.25. The last one it's going to be 37.5 X and 62.5 Y. Remember for 100 of them, equaling 1300 and 3750. Now let's call those equations 1-2 and three. A helpful hint is if one of the variables is missing, that's the variable to get rid of in the other two equations. Because if I use equation two and three to get rid of Z, that new equation 4I can then combine directly with three because three only had two variables. So let's see, I'm going to do some simplification and writing things as equivalents. The first thing I'm going to realize is that entire top equation is all divisible by 5. So I'm going to say 75 / 5, which is going to give me 15 X plus y + 4 Z equaling 2195 point 4 / 5 four 39 point O eight. It's just going to give me smaller numbers, which is going to make them a little easier to work with. For this next equation, I'm going to do a couple of things. First, I don't like the 12.5 and the Z because I'm going to combine it with a four Z. So I'm going to multiply everything through by 100 or by 10 to start. So I'm going to get 750X plus 125Y plus 125 Z equaling 22252.5. Now from there I'm going to realize that all three of those, all four of those numbers, I'm going to be able to divide by 125. So if I divide each of these by 125, that's going to give me this as six X + y + C equaling 22252 point 5 / 100 and 25178 O 2. So this is my new equivalent one, and after two different manipulations, that's my new equivalent 2. If I want to be getting rid of the Z's, I'm going to take equation 2 and multiply it through by a -4. So if I multiply it all through by -4 negative 24 X -4 Y -4 Z equal -712 O 8. So my new equation 4 is going to be, let's see, 2415 plus -24 is going to give us -9 X -3 Y, the Z's canceled 439.08 minus or plus a -700 and 12:08 it's going to give me 273 negative. Now if I look there, I can see that all of those terms are actually divisible by three. So negative three X -, y equaling -91. So that's an equivalent for equation 4. Now I want to combine with equation 3. So if we go back to equation 3, I'm going to do some manipulation again. I'm going to multiply everything 3 by 10370. Five X + 625 Y equal 13,375. Those are all going to be divisible by 25 S 15X plus 25 Y 535. And actually those were all divisible by 125 and I didn't realize it. So I'm going to take that equation and I'm going to divide it by 5 again, just trying to get things as small as I possibly can. So if I just double check that really quick on the calculator, 351 O 7 Good. So this is an equivalent to equation 3. Now I want to combine three and four. Oh, look, three and four are going to have the same coefficients. Now that's wonderful if we use the reduced forms. So now when we add these, the X's are going to cancel. I'm going to get 4Y equaling 107 -, 9116. So Y is 4. If I have Yi can now substitute it back into either equation three or four. Let's just put it in three and I get three X + 5 * 4 equaling one O 7 or 3X equaling one O 7 - 20, so 87 or X equal 87 / 329. Now, if I have Y&X, I need to find Z and I want to put it in as most simplified that I can find. So I might choose this one here because it's got AZL by itself. So if we substitute there, we're going to have six. Let's see, I'm running out of room. 6 * 29 plus Y which was 4 + C equal $178.02. So 6 * 29 is 100 and 74174 + 4 is 178. If I subtract 178 from each side, I'm going to get $0.02. So 29, four and two are our solutions for this problem.