Prices of Fast Food Use the information given in Problem 83...

Prices of Fast Food Use the information given in Problem 83 . Suppose that a third group purchased 3 deluxe hamburgers, 2 large fries, and 4 large colas for $\$ 10.95 .$ Now is there sufficient information to determine the price of each food item? If so, determine each price.

Key Concepts

Systems of Linear Equations

This concept involves setting up and solving multiple equations that contain the same set of unknowns, as each purchase provides an equation relating the prices of the items. It is a foundational tool in algebra for determining individual quantities when given aggregate data.

Determinacy and Indeterminacy

Determinate systems have a unique solution when there are as many independent equations as unknowns, while indeterminate systems do not yield unique values. Determining whether the set of equations from purchases leads to a unique solution is key in finding each food item’s price.

Substitution and Elimination Methods

These algebraic techniques are used to solve systems of equations. Substitution involves solving one equation for a variable and replacing it in another equation, while elimination involves adding or subtracting equations to cancel out a variable, both of which are essential for isolating and identifying individual prices.

Cost Combination Problems

This refers to problems where the total cost of several items purchased together is given, and the task is to determine the price of each individual item. It requires forming equations based on the combinations of items and solving for the unknown prices.

Transcript

00:01 So in this problem, where you can use the previous problems information, where we were given these orders, right, of fast food.

00:15 Hamburgers, fries, colas, and the total they paid.

00:27 And the first group got eight hamburgers, six fries, six colas, paid, $2610.

00:39 The second group got 10 hamburgers, six fries, eight colas, and they paid 3160.

00:56 And now we have a third group, and this group buys three hamburgers, two fries, four colas, and they pay $10 .95.

01:20 And the question is, is there sufficient information? and we say yes, there is sufficient information.

01:42 As we have three unknowns, right? we have hamburgers, fries, colas, the amounts, the prices on each.

02:00 This is going to be the prices.

02:03 Okay.

02:05 And we have three equations.

02:13 And as it turns out, fortunately, none of these equations are dependent on the other one, or none of them are multiple of the other one.

02:23 All right, so then it asks us to determine the prices if we can.

02:28 And so we begin to write the equations, based on this information.

02:34 We have 8h plus 6f plus 6c is equal to 2610.

02:44 10h plus 6f plus 8c is 3160 and 3h plus 2f plus 4c is 1095.

03:07 I'm going to call this equation 1, this equation 2, and this equation 3 now.

03:15 Okay.

03:18 So then if i take equation 1, minus equation 1, minus equation.

03:21 Equation 2 because when i do the fs will cancel out i get minus 2h minus 2c is equal to a negative 5 .50 and i'm going to call this equation 4.

03:43 I'll draw a line right here just to kind of keep things clear in what we're doing here.

03:48 Okay so then i got to work with equation 3 and get rid of the f.

03:54 Well i can take 3 or negative 3 times equation 3 and add it to equation 1 and the f's will disappear.

04:04 So let's do that.

04:06 Minus 3 times equation 3.

04:10 That's minus 9h, minus 6f, minus 12c equals negative 30 to 85.

04:25 And then we're going to add equation 1 to this.

04:30 So i'm just going to write it here for clarity.

04:36 6c is equal to 2610.

04:40 Add these together.

04:43 The fs cancel out.

04:46 And i'm left with minus h.

04:49 Minus 6c is equal to minus 6 .75...

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