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# Reading: Constructing and Solving Linear Equations

In practice, there are often many demands that must be met in order for the outcome of a situation to be considered a "success." In mathematics, we like to be able to quantify those demands. Consider the following situation:

### Example 1

A small business is considering purchasing bus passes for its employees in an effort to "go green." The company has 30 employees. Valley Metro sells month-long passes for $20 and half-month-long passes for$13. Assuming that a pass should be available to each employee, and that some employees will only take the bus occasionally, how many of each pass should be purchased in order to use up the allocated budget of $530? #### Solution We should first consider what it is that we are being asked to find. The question is to find the number of each of the two types of passes the company should purchase. Suppose your boss gives you the question, "How many of each pass should we purchase?" Being put on the spot without a good answer, you might answer, "We should buy x month-long passes and y half-month-long passes." Phew! You just bought yourself some time to do a more thorough analysis. So, we know that: x = # of month-long passes y = # of half-month-long passes Now we have to understand what constraints are being placed on x and y. We need to make sure: • Each employee receives a pass • The budget is$530 and should be used up
It is important to recognize that there is no constraint on what proportion of the passes should be month-long and half-month-long. This is not important to the company in these circumstances. As analysts, it is our job to ensure that all assumptions are met! For the first assumption, we need to ensure that the total number of passes is 30: x + y = 30 Notice that we are saying: "the number of month-long passes plus the number of half-month-long passes should be 30." For our second assumption, we want to make sure that the total cost is $530: • The cost for x month-long passes is$20 times x passes, or 20x.
• The cost for y half-month-long passes is $13 times y passes, or 13y. The sum of these costs should be$530. That is, 20x + 13y = 530 Perfect! We now need to solve for x and y. We need to solve the "system" of equations: x + y = 30 20x + 13y = 530 We have used at least two different methods in our past math courses: solving by substitution and solving by elimination. We will solve by substitution. x + y = 30 → y = 30 – x Substituting the right side in for y in the second equation: 20x + 13(30 – x) = 530 20x + 390 –13x = 530 7x = 140 x = 20 To find y, we can substitute the solution for x into either equation. To demonstrate that both provide the same value, we consider both scenarios:
 Equation 1 Equation 2 20 + y = 30 y = 10 20(20) + 13 y = 530400 + 13 y = 530 13 y = 130 y = 10
We conclude that y = 10 and that x = 20. This means that the company should purchase 20 month-long passes and 10 half-month-long passes in order to have a total of 30 passes and to use up the available funds exactly (note that 20(20) + 13(10) = 530). Do you feel that this is a reasonable situation? Maybe, but maybe not; however, it is important to note that the outcome is based on the assumptions. We did use up all the money and we did buy 30 passes. Then, what could be more reasonable? Maybe we should not purchase 20 of one and 10 of the other. What if all employees wanted to ride the bus occasionally? Then perhaps it would make more sense to save the extra money and purchase 30 half-month-long passes.

# Practical Considerations

• In mathematics, we always make assumptions (as we do in real life). Our solutions are "constrained" by our assumptions.
• It is not the mathematics that doesn't make practical sense, but perhaps the poor assumptions used in the process.
That leads us to our definition:

## System of Linear Equations

A system of linear equations is a collection of one or more equations that can be written in the form ax + by = c. By linear, we mean that each of the variables has a power of 1. If any of the variables have a power of more than 1, then we call the system nonlinear. We can generically write a linear equation with variables in the form: a1x1 + a2x2 + ... anxn = c where we use subscripts just to distinguish the terms as being separate. Additionally, it is not impossible to have more variables than the number of letters in the alphabet, thus by using this form we are allowed to have the number of terms be extremely large. You probably noticed that to solve our system in the example above, we had to utilize both equations. Not coincidentally, we also had two variables. You can probably see that x + y = 30 cannot be "solved" in the practical sense by itself. Certainly, we can solve for y = 30 – x, but we still know nothing about the numerical values of x! In general, if you have n variables, then you need at least n equations. It is possible to have more, but often results in no solution.

## Condition for Solving a System of Linear Equations

In order to solve a system of linear equations with variables, at least n equations are needed.

### Example 2

In reference to the first example, suppose that ticket prices go up to $21 and$13.50 for the month-long and half-month long bus passes, respectively. It is still critical to purchase 30 passes. Under the same assumptions, how many of each bus pass should be purchased?

#### Solution

We modify the second equation to match the price adjustment x + y = 30 21x + 13.5y = 530 Solving by substitution, 21x + 13.5(30 – x) = 530 21x + 405 – 13.5x = 530 7.5x = 125 x ≈ 16.7 Solving for y: 16.7 +y = 30 y = 13.3 We have solved for x and y, but we should note that it is not practical to purchase fractional tickets. Since it is critical to purchase 30 tickets, we consider the two scenarios below:
 x y Total Cost 17 13 21(17) + 13.5(13) = $532.5 16 14 21(16) + 13.5(14) =$525

Clearly, x = 17 and y = 13 passes is over budget. While not ideal, the company should probably settle for x = 16 month-long passes and y = 14 half-month-long passes in order to meet its budget constraint. It will end up $5 under budget. As we well know, these equations can be graphed. When graphing, it is a good idea to find the vertical and horizontal intercepts to get an idea of the domain and range of the function. This exercise will be left to the reader. As we can see, the point where all involved equations are made true by a single set of values represents the intersection point of their graphs. ## Solution to a System of Linear Equations The solution to a system of linear equations involving variables is the value set (x1, x2,...xn), such that it makes all equations true. Graphically, this is the intersection point of the graphs of these equation Note: It is possible to have, for example, 50 variables. Thus, the intersection point (if it even exists) would be a set of 50 ordered numerical values. ### Example 3 Determine whether the point (w,x,y,z) = (1,4,2,0) is a solution to the following system of linear equations: w + x + y + z = 7 2w + z = y xz + 3y = 10 #### Solution We must test the point to see whether it satisfies the three equations given (notice that there are four variables but only three equations. This is okay, since we are not solving the system, rather testing to see if the point represents a solution). w + x + y + z = 7 1 + 4 + 2 + 0 = 7 7 = 7; TRUE 2w + z = y 2(1) + 0 = 2 2 = 2; TRUE xz + 3y = 10 4 – 0 + 3(2) = 10 10 = 10; TRUE Since the point satisfies all three equations, is a valid solution to the system of linear equations. Often times, the difficulty with systems of equations is not necessarily found in solving them, but in actually writing them! Here are a few steps that seem to work quite well. ## To Model a Real-World Situation with a System of Equations 1. Locate the question. Answer the question by stating, "purchase x units of ________, y units of _________, …" This is a natural way to define your variables, which are the unknown quantities you want to determine. 2. Define your variables explicitly, e.g. x = the # of units of iPods, y = # of units of iPads 3. Read through the problem again. In words, write out any conditions that are necessary as bullet points. For example, "They expect to sell twice as many iPods as iPads." 4. Convert your verbal expression into mathematics. E.g. x = 2y 5. Solve the system. 6. Interpret your answer and assess whether your answer makes sense. Hint: It is often confusing for us to read statements such as, "Tom's Auto will order one-third as many trucks as cars for his lot." Before forfeiting to intimidation, try the statement both ways. First, let's say t = # of trucks ordered; c = # of cars ordered So, is $\displaystyle{c}=\frac{{1}}{{3}}{t}$ cars. But, he actually gets fewer trucks than cars! If we try, $\displaystyle{t}=\frac{{1}}{{3}}{c}$ trucks, which is the way we want it. Punchline: Spend the time to give yourself an example prior to taking a wild guess! ### I Bet You're Wondering . . . #### Question How do we know that the number of managers in each of the two groups is 597 and 118? #### Answer We really don't. With the given information, we are using past data to give us a glimpse into what is required. That is, to get exactly enough votes, there would need to be this split. # Practice Problems 1. Between 2005 and 2009, the percentage of people who own a home has changed drastically from state to state. The percentage of individuals who own a home in Indiana can be modeled by C(t) = –0.6t + 76, where t is the number of years since 2005. For New Hampshire, the same model is given by N(t) = 0.5t + 73. Did or will Indiana and New Hampshire ever have the same homeownership rate? (Source: Modeled from U.S. Statistical Abstract, Table 988) Solution: 2. Milos is an avid maker of tinctures, which are concentrated liquid forms of natural herbs. One common herb that is formed into a tincture is the vanilla bean, which is sold in stores under the moniker "Vanilla Extract." One means by which to create a tincture is by soaking the herb in grain alcohol for several weeks. It is possible to purchase grain alcohol called Everclear, which is 97% alcohol; However, vanilla requires a 60% alcohol solution (97% is too strong). To adjust for alcohol that is too strong, the solution can be diluted with water (0% alcohol) to achieve the desired concentration. Suppose Milos is planning on making a 12-ounce vanilla extract. How much water and alcohol should he use?Solution: 3. In the world of finances, you will often hear of companies choosing to "hedge" their risk. In simple terms, this means that a company can offset risk by taking some action. For example, when companies have debt to repay at some time in the future they will often invest a smaller sum of money so that it accumulates to the amount of the debt when that amount is due. In order to avoid paying capital gain taxes on any amount of money earned above that which is due, the company will plan to earn the exact amount of debt that is due in the future.Solution: TIAA-CREF, a Fortune 100 financial services company has a wide range of variable annuity accounts. A few of those are listed below: Type 10-Year Average Annual Return Global Equities 3.61% Social Choice 4.51% Bond Market 5.29% Money Market 1.97% SOURCE: http://www.tiaa-cref.org/public/performance/retirement/index.html 1. A company has$10,000 in liabilities that are due in one year. It chooses to invest in the Bond Market and Social Choice accounts. The company would like to earn $500 in interest so that they are only investing$9,500 of their own funds. Assuming the 10-year average annual return, how much should they put in each of the two accounts?
2. How much should they invest in each if they decide to go with the Global Equities and Money Market accounts and are satisfied with $250 (they will invest the rest) in interest assuming the 10-year annual average? 3. Why would the company be in over its head if it attempted to make$1,000 in interest in one year from with the investments in the table above as its choices? Provide mathematical evidence to justify your answer.