Library Makerspace: Tools, Kits, and Systems

Use inspection, algebra, and graphs to analyze lending records.

Inspect the Records

These systems test the library's lending software. Signed values are allowed. Use inspection when possible.

Two checkout counters use the rules y = x + 4 and y = 4 − x. Find the values that make both rules true.

x = ; y =

Two records claim that 3x + y = 12 and 3x + y = 18 describe the same inventory. What can you tell by inspection?

A. The system has infinitely many solutions.B. The system has the solution (4, 6).C. The system has the solution (2, 6).D. The system has no solution.

A software check uses y = 5x + 2 and 3y = 15x + 6. Which statement describes all solutions?

A. Only the point (0, 2) solves the system.B. Every point on y = 5x + 2 solves the system.C. No point on y = 5x + 2 solves the system.D. Every point in the coordinate plane solves the system.

A lending-record system is 2x + 5y = 19 and 3x − 2y = 0. Which first step would let you eliminate x by subtraction?

A. Multiply the first equation by 3 and the second by 3.B. Multiply the first equation by 2 and the second by 3.C. Multiply the first equation by 3 and the second by 2.D. Multiply the first equation by 2 and the second by 2.

Solve and Apply

Solve each system using algebra. Show your steps on separate paper. Each inventory situation is independent.

A checkout test uses x = y + 6 and 2x + 3y = 27. Solve by substitution.

x = ; y =

Two tool-tracking rules give 4x + 3y = 11 and 2x − 3y = 1. Solve by elimination.

x = ; y =

Signed adjustments in a software test satisfy 3x + 2y = 1 and 5x + 4y = 0. Find both adjustments.

x = ; y =

A kit-label calibration uses 0.5x + y = 4 and 2x − y = 1. Solve the system.

x = ; y =

The library lends 17 kits in one afternoon. Each basic kit contains 2 tools, and each advanced kit contains 5 tools. The kits contain 58 tools altogether. Let x be the number of basic kits and y be the number of advanced kits. Write a system on separate paper, then solve it.

Basic kits: ; advanced kits:

10.

The makerspace buys replacement hand tools and empty kit cases. Each tool has the same price, and each case has the same price on both orders. The totals exclude tax. Use the records to find each item's price.

Replacement Orders
Order	Hand tools	Kit cases	Total cost
First	3	2	$46
Second	2	3	$54

One hand tool costs $; one kit case costs $.

11.

Before checkout, a cabinet has x measuring kits and y fastening kits. After 7 measuring kits and 2 fastening kits are borrowed, equal numbers of the two types remain. An inventory check also gives 2x + y = 46. Write the equal-remaining-counts equation, then solve the system.

Originally, there were measuring kits and fastening kits.

Graph and Interpret

Use graphs to estimate intersections. Round requested coordinates to the nearest tenth. For the final question, explain what the algebra means.

12.

The library's kit-label layout uses signed coordinates. The graph shows one guide line, y = (1/2)x − 2. Sketch the second guide line, y = −x + 2, on the same grid. Which ordered pair best estimates where the guides cross?

A. (2.7, −0.7)B. (2.7, 0.7)C. (−0.7, 2.7)D. (−2.7, −0.7)

13.

Two guide lines on another kit label follow y = 2x + 3 and y = −x − 2. Graph both lines through at least two points each. Mark their intersection and label its approximate coordinates to the nearest tenth.

14.

An inventory sheet gives 2x + 3y = 24, where x and y count two types of lending kits. A second sheet gives 4x + 6y = 48. When Rowan subtracts twice the first equation from the second, the result is 0 = 0. Rowan says, “Two equations must give us the exact number of each kit.” Explain why these records do not determine one inventory. Give two different nonnegative whole-number pairs (x, y) that fit both records.