A health insurance company advertises on television, on radio, and in the local newspaper. The m

A health insurance company advertises on television, on...

Key Concepts

Substitution Method

The substitution method is a strategy for solving systems of equations where one variable is expressed in terms of another from one equation, and this expression is substituted into the remaining equations. This reduces the number of variables and simplifies the process of finding a solution.

Constraint Equations

Constraint equations are mathematical expressions that encapsulate the limitations or requirements of a problem. These may include resource limits, total counts, or specific relationships between variables that must be satisfied for a solution to be feasible.

Modeling with Variables

Introducing variables allows us to abstract and represent unknown quantities in a problem. By assigning symbols to these unknowns, we can convert real-world scenarios into mathematical equations that can be systematically analyzed and solved.

System of Linear Equations

This concept involves setting up and solving multiple equations simultaneously, each representing a constraint or relationship between the variables in a problem. It is widely used in many disciplines to model situations where several conditions must hold at the same time.

Transcript

00:01 So for this problem, it's a word problem, which i know everybody loves, but part of the difficulty of it is setting it up because if you don't set it up correctly, then obviously you can't solve it correctly.

00:13 So i'm going to walk through how to set this one up.

00:15 So first off, we should define our variables.

00:18 So i'm going to type out here that we've got t and i'm going to say that that's, oops, t is going to be television ads per month.

00:32 We've got r.

00:34 That's going to be radio ads per month.

00:39 And we've got n, and that's going to be newspaper, ads per month.

00:46 Okay, so we have a couple of constraints given to us.

00:50 One of them is about cost.

00:51 One of them is about the number of ads, and one of them is about the proportion of the ads.

00:55 So we have three variables, and we're tasked with solving all of them.

01:01 So we need three different equations.

01:03 Remember, your number of equations always needs to match or exceed the number of unknowns you have.

01:10 So the first one we're going to set up is about cost.

01:12 They told us that we have a budget of $42 ,000, and they told us how much each ad costs, or each type of ad.

01:20 So i said it cost $1 ,000 for every television ad.

01:24 And then it's said we have $200 for every radio ad, and then $500 for every newspaper ad and that our total budget was $42 ,000.

01:37 Okay, and then in the next equation, we just know that we have 60 ad slots per month.

01:44 And so all of these are just numbers of ads that we don't know.

01:50 So i'm going to go ahead and add all of them together.

01:54 So the number of tv, radio and newspaper ads equal 60.

02:00 And then we have the fact that the the radio and newspaper ads combined equal the number of television ads.

02:13 Okay.

02:13 So now what we're going to do is we're going to try and solve this system of equations.

02:16 We have three equations in this system.

02:19 And whenever i see something like this where it defines one variable in terms of another, it's really easy for us to go back and substitute some things back in for these variables.

02:31 So i see r plus n equals t.

02:34 So that tells me anywhere that i see t, i can now replace it with r plus n.

02:40 So we're going to go ahead and do that below.

02:42 So i know that t equals r plus n.

02:45 So i'm going to substitute that.

02:46 So now i've got 1 ,000, not times t, but times r plus n.

02:52 And nothing else changes.

02:54 I'm just substituting what i know t is.

02:57 Because i don't know the value.

02:58 I do know the relationship.

03:01 But not all the rest of it changes.

03:03 Okay.