Linear equations with one, none or infinitely many solutions Transitional Reasoning Practice

Questions

1. 1.

   A water management facility monitors two reserve tanks during an automated refill process. The volume of water (in liters) in Tank A after $t$ minutes is modeled by $V_A = 6(2t + 15) - 2t$, while the volume in Tank B is modeled by $V_B = 5(2t + 20) - 8$. An engineer sets up the equation $V_A = V_B$ to determine the time $t$ at which both tanks contain the same volume of water. Which of the following statements correctly describes the solution to this equation and its operational meaning?

   • A) The equation has exactly one solution at $t = 1$ minute, meaning the tanks reach equal volume after 1 minute of refilling.
   • B) The equation has infinitely many solutions, meaning the two tanks maintain identical water volumes at every minute of the process.
   • C) The equation has exactly one solution at $t = 2$ minutes, meaning Tank B catches up to Tank A after 2 minutes.
   • D) The equation has no solution, meaning the two tanks will never contain the same volume of water at any time during the process.
2. 2.

   A hydraulic engineer models the equilibrium flow rate $x$ of a bypass valve using the linear equation $m(x - 2) + 4x = 2(3x - 1) + n$, where $m$ and $n$ are constant control parameters. The team needs to determine how specific combinations of $m$ and $n$ affect the number of valid operating points for the system. Based on this equation, which of the following statements is correct?

   • A) If $m = 2$ and $n \neq -2$, the equation has a unique solution.
   • B) If $m \neq 2$, the equation has no solution regardless of the value of $n$.
   • C) If $m = 2$ and $n = -2$, the equation has infinitely many solutions.
   • D) If $m \neq 2$ and $n = -2$, the equation has no solution.
3. 3.

   An electrical engineer configures two automated power-monitoring circuits whose readings in milliwatts, as a function of system load $x$, are modeled by $P_1 = 6(2x - 1) - 5x$ and $P_2 = a(x + 2) + b$. To ensure seamless redundant calibration, the engineer requires the equation $P_1 = P_2$ to have infinitely many solutions so that both circuits report identical readings across all system loads. If the parameter $a$ is set to satisfy this condition, what must be the value of the constant $b$?

   • A) $-6$
   • B) $-8$
   • C) $8$
   • D) $-20$
4. 4.

   A robotics technician programs two automated inspection carts to move along a linear track, where their distances from the base station in meters after $t$ seconds are modeled by $d_1 = 5(2t + 3) - 4t$ and $d_2 = m(t + 1) + 7$. The technician wants to set the calibration parameter $m$ so that the carts never occupy the same position at any time $t$. Which value of $m$ ensures that the equation $d_1 = d_2$ has no solution?

   • A) $m = 10$
   • B) $m = 8$
   • C) $m = 6$
   • D) $m = 4$

Answers and step-by-step solutions appear after you finish the test.