MA, Stanford University
Teaching in the San Francisco Bay Area
Alissa is currently a teacher in the San Francisco Bay Area and Brightstorm users love her clear, concise explanations of tough concepts
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MA, Stanford University
Teaching in the San Francisco Bay Area
Alissa is currently a teacher in the San Francisco Bay Area and Brightstorm users love her clear, concise explanations of tough concepts
This is a really typical coin problem that's like what you're going to see in your early homework problems. Carl has 28 coins on his desk that are all dimes and quarters. Altogether, the coins are worth $5.65. How many quarters does he have?
Well a lot of students might approach this problem by making a guess and check chart. Here is what that would look like. You pick some numbers for quarters, you pick some number for dimes, you calculate the value and you see if it's equal to $5.65 or not. This can get annoying; let me show you what I mean.
Let's just say I guessed 10 quarters and then 18 dimes. I'd have to sit there and figure out how much 10 quarters is worth, how much 10 dimes is worth, add them together and check and I'm telling you right now, this is the incorrect answer I would have to go and check again, and check again. This takes all day guys and that's one of the things I like about Math is there's almost always a short cut especially once you get into Algebra class.
So instead of setting up a guess and check chart, a system of equations will tell me the exact answer the first time around. I'm going to want to write one equation that represents the amount of coins. My other equation is going to represent the value of those coins. Let's do the amounts first.
Number of quarters plus number of dimes is equal to 28. That I got from the very first sentence. Now I need to do something with the values. The value of a quarter is 25 cents, so .25 times the number of quarters plus.10 times the number of dimes is going to give me Carl's total value, $5.65, 565, that's all I need to do for the set of my equations, my system of equations is ready to solve. I have 2 equations with two unknowns.
So from here you can choose if you want to solve by graphing, substitution, elimination or maybe matrices if you learnt that in your classes, totally up to you. The way I decide is by looking at the coefficients in front of my variables and seeing if I have any coefficients that are 1. If I do, then I'm going to want to use substitution.
Like in this first equation, I could solve this equation easily for either quarters or dimes because they both have coefficients of 1. I'm going to solve this guy for dimes. I'll say dimes is equal to 28 take away quarters. Now what I'm going to do is substitute that 28 minus q business down here where I had a d. I'll rewrite the second equation only instead add d right there, I'm going to write 28 take away q. .25q plus .10 times my old d which I'm now going to represent by this using substitution is equal to $5.65.
Go through and solve. Make sure you distribute, you be careful with your decimal points. .25q plus that's going to be 2.8 minus .1q is equal to $ 5.65 cents. Combine your Qs .25q take away .1q is going to be .15q plus 2.8 is equal to 5.65. Then subtract 2.8 from both sides, so I'll have .15q is equal to 5.65 take away 2.8 gives me 2.85. Then I need to divide both sides by .15 and I'm going to get that I have 19 quarters, q equals 19.
After all of this business, I found out how many quarters they asked for. If I wanted to, I could go back and find out how many dimes I got from this problem or Carl has, but I don't have to. I'm done, it says how many quarters does he have. If I trust myself, I could just circle that answer and move on. If I'm not totally sure and if you're like me you want to make absolutely sure you had the problems correct before you turn them in, or if you were on a test and there's no reason not to check your work, what I would do is continue by saying if he has 19 quarters and his quarters plus dimes is 28, that means he's going to have 28 take away 19, he's going to have 9 dimes, and then I would go through and check that in the second equation and make sure I got it correct.
So it's totally up to you. You guys always have the choice whether or not you want to check your work if the problem doesn't specify, or you guys when checking is not so hard why would you not check your work? It's a way you can be sure you're getting every problem correct before you go into class. Or if you're finding you're getting problem incorrect by checking your work, you'll be able to have specific questions to ask your teacher when you show up for tutoring hours.
Unit
Word Problems Using Systems of Equations