The following table gives the number of people (in...

The following table gives the number of people (in thousands) who visited Australia and South Africa in $1998^{6}$ : $$ \begin{array}{|l|c|c|} \hline & \text { To } & \text { Australia } & \text { South Africa } \\ \hline \text { From } & \text { North America } & 440 & 190 \\ \hline \text { Europe } & 950 & 950 \\ \hline \text { Asia } & 1,790 & 200 \\ \hline \end{array} $$ You predict that in $2008,20,000$ fewer people from North America will visit Australia and 40,000 more will visit South Africa, 50,000 more people from Europe will visit each of Australia and South Africa, and 100,000 more people from Asia will visit South Africa, but there will be no change in the number visiting Australia. a. Use matrix algebra to predict the number of visitors from the three regions to Australia and South Africa in 2008 . b. Take $A$ to be the $3 \times 2$ matrix whose entries are the 1998 tourism figures and take $B$ to be the $3 \times 2$ matrix whose entries are the 2008 tourism figures. Give a formula (in terms of $A$ and $B$ ) that predicts the average of the numbers of visitors from the three regions to Australia and South Africa in 1998 and $2008 .$ Compute its value.

The following table gives the number of people (in thousands) who visited Australia and South Africa in $1998^{6}$ :
$$
\begin{array}{|l|c|c|}
\hline & \text { To } & \text { Australia } & \text { South Africa } \\
\hline \text { From } & \text { North America } & 440 & 190 \\
\hline \text { Europe } & 950 & 950 \\
\hline \text { Asia } & 1,790 & 200 \\
\hline
\end{array}
$$
You predict that in $2008,20,000$ fewer people from North America will visit Australia and 40,000 more will visit South Africa, 50,000 more people from Europe will visit each of Australia and South Africa, and 100,000 more people from Asia will visit South Africa, but there will be no change in the number visiting Australia.
a. Use matrix algebra to predict the number of visitors from the three regions to Australia and South Africa in 2008 .
b. Take $A$ to be the $3 \times 2$ matrix whose entries are the 1998
tourism figures and take $B$ to be the $3 \times 2$ matrix whose entries are the 2008 tourism figures. Give a formula (in terms of $A$ and $B$ ) that predicts the average of the numbers of visitors from the three regions to Australia and South Africa in 1998 and $2008 .$ Compute its value.

Key Concepts

Scalar Multiplication

Scalar multiplication involves multiplying every element of a matrix by a scalar (a constant number). This operation is crucial when uniformly scaling a matrix to compute averages or to apply proportional adjustments across all entries.

Matrix Addition

Matrix addition is the process of adding two matrices by combining corresponding entries. It is a fundamental operation in matrix algebra used to integrate adjustments or changes to a data set, such as updating original figures with predicted variations.

Matrix Averaging

Matrix averaging combines two matrices by adding them together and then multiplying the result by a scalar (typically one-half). This method is used to find an average or midpoint value for corresponding entries from two distinct data sets, facilitating comparison and analysis over time.

Matrix Representation

This concept involves organizing data into rows and columns in a structured manner using a matrix format. It enables the compact representation of data, such as tourism figures or any multi-dimensional information, which can then be manipulated algebraically.

Matrix Transformation

Matrix transformation refers to applying operations, such as addition, subtraction, or multiplication by constants, to change or update the values in a matrix. In forecasting scenarios, transformations represent the modifications that occur over time or between different conditions.

Transcript

00:01 For this problem, we're going to be offering some extensive, an extensive view of how to solve two equations when you have two unknown variables.

00:11 So really the possibilities are endless with this kind of stuff.

00:15 Sometimes you're talking about foods, sometimes you're talking about costs and percentages.

00:23 But we see that this comes up a lot in math where we have two different equations and two unknowns.

00:28 And because of that, because as long as there are linear equations, we can find solutions, or we can show that there's no solution.

00:39 So what this looks like is we have some values, maybe 3x plus 2y equals 5.

00:48 So off the top of my head, something this could represent is that you have hamburgers and you have french fries.

00:57 X is hamburgers, y is french rice.

01:00 You know, the hamburgers cost $3 and the french rice cost $2.

01:04 I know super unrealistic, but what you know is that after this order, you ended up getting, it ended up costing $5.

01:16 So then another, we would need another equation because we have 3x plus 2y equals 5.

01:23 So we would need another function here to represent something else.

01:32 So perhaps you know that x plus y.

01:37 So this is just the hamburgers plus the french fries.

01:41 You know that combined there's two items.

01:44 Well, what that tells you is that the number of hamburgers purchased, there is one hamburger purchased, and there was one french fry purchased.

01:53 However, this can become much more complicated.

01:55 We could increase the prices and all this kind of stuff.

02:01 And these lines would shift around, but they would still have at most one point of intersection.

02:08 They could have an infinite number of points of intersection if they were the same line, potentially.

02:15 So that's how we do it.

02:17 There's a couple things we want to look out for.

02:19 So when we have these two equations, we want them to be unique equations.

02:24 So for example, if i had put 6x plus 4y equals 10, notice that that's the exact same equation.

02:33 I could just divide this whole thing by 2, and i would end up getting this.

02:37 So that's why we end up getting the same equation as a result.

02:41 Another thing that we want to be careful with is making sure we have the right set of equations.

02:49 So going back, notice that i made sure that x, was hamburgers and y was french fries...

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