Robot Bench: When Do the Rules Agree?

Simplify equations to identify one solution, no solution, or infinitely many solutions.

Simplify the Bench Notes

The school robotics club compares rules while building its robot. Treat x as a real number unless a question gives a physical restriction. Simplify before deciding how many solutions an equation has.

A motor-check note contains 12 − 3(x − 2) = 2x − 7. Which equation is equivalent after distributing and combining like terms on each side?

A. 18 − 3x = 2x − 7B. 12 − 3x = 2x − 9C. 6 − 3x = 2x − 7D. 18 + 3x = 2x − 7

For a controller check, simplify 8(x + 2) − 3x = 5x + 21. Subtract 5x from both sides, then subtract 16 from both sides.

Complete the resulting statement: 0 = .

A sensor-check equation is (1/2)(6x − 14) = 3x − 7. Which conclusion follows from simplifying it?

A. It becomes −14 = −7, so no real number is a solution.B. It becomes x = 0, so only zero is a solution.C. It becomes −7 = −7, so every real number is a solution.D. It becomes 3x = 7, so only x = 7/3 is a solution.

After simplifying a robot test equation, Priya gets 0x = 11. What does this tell her about the original equation?

A. It has no real solutions.B. It has infinitely many real solutions.C. It has exactly one solution, x = 0.D. It has exactly one solution, x = 11.

The equation 4(2x − 3) = 5x + 9 appears in a wheel-test log. It has exactly one solution.

Find that solution: x = .

Apply the Checks

Use each robot-building situation to decide when two quantities can be equal. A few matching trials alone do not prove that rules agree for every input.

Two proposed support rails have lengths 3(x + 4) centimeters and 3x + 15 centimeters. The adjustable length x must be positive. Can the club choose x so the rails have equal lengths?

A. Yes; they are equal only when x is 1 centimeter.B. No; the first rail is always 3 centimeters longer.C. Yes; they are equal for every positive value of x.D. No; the second rail is always 3 centimeters longer.

During a straight-line test, one wheel turns at 5 revolutions per second. The other turns at 4 revolutions per second and starts 6 revolutions ahead. Their total revolution counts agree when 5x = 4x + 6, where x is elapsed time in seconds. Which conclusion is correct?

A. The counts never agree during the test.B. The counts agree only at 30 seconds.C. The counts agree at every elapsed time.D. The counts agree only at 6 seconds.

The club plans two identical wire runs. Each run uses x centimeters plus 9 centimeters for slack. One student records the total as 2(x + 9); another records x + x + 18. For how many positive values of x do the totals agree?

A. They agree for every positive value of x.B. They agree only for x = 18.C. They agree only for x = 9.D. They agree for no positive value of x.

Two distance sensors apply different corrections to the same raw reading x. Their corrected readings are shown below.

Sensor Correction Rules
Sensor	Corrected reading (cm)
Front	0.5(x + 8)
Rear	0.25x + 7

Set the two expressions equal. The equation has exactly one solution.

At what raw reading do the corrected readings agree? cm

10.

A diagnostic screen compares 6x + 2 with 2(3x + 1). A club member wants to test every possible input. Which simplification makes that unnecessary?

A. The equation reduces to 2 = 1, so no input works.B. The equation reduces to 2 = 2, so every input works.C. The equation reduces to x = 2, so just one input works.D. The equation reduces to x = 0, so just zero works.

Design and Audit Examples

Complete examples and check the club's reasoning. Use algebra, not just trial values.

11.

A test worksheet starts with 9x − 4 = ___. Which right-hand expression makes an equation with no solution?

A. 8x − 4B. 9x − 4C. 9x + 4D. 10x + 4

12.

The club needs examples that are true for every real input. Select every equation with infinitely many solutions.

13.

Sam writes three equations for a robot diagnostic lesson. Which set correctly gives one example of each solution type?

Sam's Equations
Name	Equation
Drive	5(x − 1) = 5x − 5
Grip	5(x − 1) = 5x − 2
Turn	5(x − 1) = 4x − 2

A. Drive: one; Grip: none; Turn: infinitely many.B. Drive: infinitely many; Grip: none; Turn: one.C. Drive: infinitely many; Grip: one; Turn: none.D. Drive: none; Grip: infinitely many; Turn: one.

14.

In a controller check, Mateo simplifies −2(x − 5) + x = 10 − x. He writes −2x − 10 + x = 10 − x, then concludes that −10 = 10 means the original equation has no solution. Identify his error. Simplify the original equation correctly and explain its actual number of solutions.