Solutions of Linear Equations

  1. For each of these linear equations, find the value(s) of \(x\) that satisfy it.
    \(\displaystyle 7x-17=23-2x\)
    \(x = 40/9 = 4.\overline{4}\)
    \(\displaystyle 1.41x = 2.71x - 3.14\)
    \(x \approx 2.4153846\)
    \(\displaystyle \frac{5(x-2)}{3} - 1 = \frac{x}{3} + \frac{1}{5} \)
    \(x = 3.4\)
    \(\displaystyle \frac{3-x}{2} = \frac{1}{3} + 7x\)
    \(x = 7/45 = 0.1\overline{5}\)
  2. Solve the equation \( y = mx + b \) for \(m\).
  3. Solve the equation \( 4.1(3\varphi - 1.6) = 1 \) for \(\varphi\).
  4. Solve the equation \( 2y^2 -3x = 5(x - 9k) \) for \(x\).
  5. For an investment of \(P\) dollars growing at an annual interest rate of \(r\) that is compounded monthly, the following formula describes the value \(S\) of the investment after one month. \[ S = P\left(1 + \frac{r}{12}\right) \]
    1. Solve the formula for \(P\).
    2. Solve the formula for \(r\).
    3. If you have $10,000 and would like to invest it to earn $17 in interest after a single month, what annual interest rate should you look for?
  6. The price of Johnson & Johnson (NYSE:JNJ) stock closed at $158.48 per share on 14 August 2024, but closed at $137.49 per share on 29 November 2019. What was the average monthly change in share price between these two dates? If the share price keeps increasing at this average rate, when will the price per share reach $200?

  7. The American Heart Association recommends that when exercising, a person should keep their heart rate, measured in beats-per-minute (bpm), within a specified range for optimum health; too low and the exercise won’t have any benefit, too high and the exercise is too strenuous. They call this range the Target HR Zone, and their recommendation varies depending on a person’s age according to the following table.

    Age (years) 20 30 40 50 60 70
    Bottom of Target HR Zone (bpm) 100 95 90 85 80 75
    Top of Target HR Zone (bpm) 170 161.5 153 144.5 136 127.5
    1. Both the top and bottom of their recommended target heart rate zone appear to vary linearly with a person’s age. What are the formulas of the two linear functions that determine the top and bottom of this target hear rate zone as a function of age?
    2. Come up with a linear function that takes a person’s age as input and returns the heart rate that is exactly in the middle of the target heart rate zone for that person.

Some Challenges

  1. What is an equation for the line that passes through the point \((22,-6)\) and whose \(x\)-intercept is five more than its \(y\)-intercept?
  2. Recall that a rhombus is a four-sided shape characterized by the fact that each pair of opposite sides are parallel. If the points \((3,6)\) and \((-1,2)\) are the coordinates of opposite vertices of a rhombus in the plane, what’s an equation for the line that passes through the other two vertices?