Systems of Linear Equations and Inequalities

  1. Janet has $123,000 she’d like to invest. She’s researched two different investments accounts:
    • A fund offered by her local credit union that guarantees a 6.9% annual return.
    • A high risk mutual fund that is projected to net 7.7% annual return.
    She wants to balance her investment between these two funds to get an expected return of 7.2%. How should should split up her principal investment between these two funds?
    Solution Letting \(A\) be the amount she invests in the money-market fund and \(B\) be the amount she invests in the mutual fund, we can write two equations: one about the total principal (how much money she has to invest), and another equation about the interest she expects. Then the answer we seek is the solution to the system \[ \begin{cases} A &+&B &=& 123000 \\0.069A &+&0.077B &=& 8865 \end{cases} \,. \] Solving this system, we calculate that she should put $76,875 in the money-market fund and $46,125 in the mutual fund.
  2. Solve this system of linear equations.
    \(\begin{cases} 3x-2y = 12 \\ 7x+y = 11 \end{cases}\)
    \(x = 2 \quad y = -3\)
  3. Solve this system of linear equations.
    \(\begin{cases} x+4y = 3 \\ -3x+5y = 1 \end{cases}\)
    \(x = 11/17 \approx 0.647 \quad y = 10/17 \approx 0.588\)
  4. What are the coordinates of the point where these two lines intersect?
    \(y = 3x-2 \quad\text{and}\quad y = \frac{1}{4}x+5\)
    \(x = 28/11 = 2.\overline{54} \quad y = 62/11 = 5.\overline{63}\)
  5. Mountain Dew™ contains 3.875g sugar per fluid ounce, whereas orange juice contains about 2.625g sugar per fluid ounce. How many fluid ounces of each would you need to mix to get a gallon (128 fl.oz.) of a delicious drink that has 3g sugar per fluid ounce?
    Solution Letting \(D\) be the number of ounces of Mountain Dew and \(J\) be the number of ounces of orange juice, we’ll write one equation relating the total amount of fluid we want, and another equation relating the total amount of sugar we want. Then the answer we seek is the solution to the system \[ \begin{cases} D &+& J &=& 128 \\ 3.875D &+& 2.625J &=& 384 \end{cases}\,. \] Solving this system, we calculate that we’ll need \(D = 38.4\) fluid ounces of Mountain Dew and \(J = 89.6\) fluid ounces of orange juice.

  6. My bro Mikey is on a very special diet of bananas and watermelons. He also holds himself to a very strict dietary requirement of getting 37.5g of fiber and 214.4g of sugar per day. (Note: this is a healthy amount of fiber, but an unhealthy amount of sugar.)

    How many bananas and cups of watermelon should Mikey eat daily to maintain his diet?
    Solution Let \(B\) be the number of bananas and let \(W\) be the number of cups of watermelon that Mikey plans to eat. Writing one equation relating to Mikey’s fiber intake and another relating to his sugar intake, the answer we seek is the solution to the system \[ \begin{cases} 2.6B &+&0.6W &=& 37.5 \\ 12.2B &+&9.3W &=& 214.4 \end{cases} \,. \] Solving this system, we calculate that he needs to eat \(B=13\) bananas and \(W=6\) cups of watermelon per day.

  7. The drugs Aryzindrid and Boquaflox were each designed to treat the rare medical condition oxyfloxridomosis. Aryzindrid just recently passed drug trials and is hitting the market, while Boquaflox has been prescribed since the mid 90s.

    At the beginning of last year, Aryzindrid only had a 1.5% share of the oxyfloxridomosis treatment market. However the manufacturers of Aryzindrid funded a huge marketing campaign and by the end of last year Aryzindrid’s market share was 62.3%!

    Boquaflox had 81.1% market share at the beginning of last year, but after the marketing push by the makers of Aryzindrid, it dropped to a 23.7% share by the end of the year.

    1. At what average rate per week did Aryzindrid gain market share last year? At what average rate per week did Boquaflox lose market share last year? (Remember there are fifty-two weeks in a year.)

      Solution \[ \frac{62.3\% - 1.5\%}{52\text{ weeks}} \approx 1.17\%\text{ per week} \qquad \frac{23.7\% - 81.1\%}{52\text{ weeks}} \approx -1.1\%\text{ per week} \]

    2. Assuming the market share of each drug changed linearly throughout the year — i.e. each function of their market shares over time can be approximated as a linear equation — what week during the year did the two drugs have the same market share?

      Solution Writing down the linear equations for each their market shares, the answer we seek is the \(t\)-coordinate of the solution to this system of equations: \[ \begin{cases} y &= 1.17t + 1.5 \\ y &= -1.1t + 81.1 \end{cases} \] (Remember the units on \(y\) are \(\%\).) Solving this system, we calculate that \(t \approx 35\), so their market shares were the same at the thirty-fifth week. At this week, \(y \approx 42.53\%\), so this was their market share at this time.