More Variables Than Equations in a System
Date: 01/21/2002 at 16:35:27 From: Penny Subject: Equations with multiple variables Zeke, Maisie and Ivan stopped at a fruit market for an afternoon snack. Zeke bought an apple and a mango, and he paid $1.40. Maisie bought a pear and an orange, and she paid 90 cents. Ivan was really hungry, so he bought a mango, an orange, a pear, and an apple. How much did Ivan pay? I'm trying to use variables, but I have four variables and there doesn't seem to be enough information to figure it all out. Can you help me?
Date: 01/21/2002 at 17:14:30 From: Doctor Riz Subject: Re: Equations with multiple variables Hi Penny - While having more variables than equations is often a difficult situation, in this particular problem you don't have to figure out how much each piece of fruit costs in order to answer the question. I agree that we could use four variables. Let's use these: a = cost of an apple m = cost of a mango o = cost of an orange p = cost of a pear We know that Zeke paid $1.40 for an apple and a mango, and Maisie paid 90 cents for a pear and an orange, so: a + m = 1.40 and p + o = 0.90 Ivan bought a mango, an orange, a pear, and an apple, so: m + o + p + a = Ivan's bill As you said, we don't have enough information to figure out each variable. But let's rearrange Ivan's equation: (a + m) + (p + o) = Ivan's bill Can you see that this is the same as his original equation? We can change the order in which we add things without changing the result. Now we can use some substitution. Since we know from the earlier equations that a + m = 1.40 and p + o = 0.90, we can substitute those into Ivan's equation to get: 1.40 + 0.90 = Ivan's bill 2.30 = Ivan's bill We can now see that Ivan would have paid $2.30 to buy one of each kind of fruit. Again, the key idea in this problem is that we did not determine the individual cost of any of the pieces of fruit, but we were still able to find the cost of buying one of each. Does that help? Feel free to write back if you need more help. - Doctor Riz, The Math Forum http://mathforum.org/dr.math/
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