A research laboratory placed two orders for high-precision sensors. The first order, consi… — GMAT

A research laboratory placed two orders for high-precision sensors. The first order, consisting of $5$ acoustic sensors and $3$ thermal sensors, cost a total of $\$1{}160$ . The second order, consisting of $3$ acoustic sensors and $5$ thermal sensors, cost a total of $\$1{}080$ . If each acoustic sensor costs the same amount and each thermal sensor costs the same amount, what is the combined cost of $1$ acoustic sensor and $1$ thermal sensor?

A
$\$140$
B
$\$160$
$\$280$ Answer
D
$\$320$
E
$\$560$

Answer

The combined cost of 1 acoustic sensor and 1 thermal sensor is

\$280

By representing the acoustic sensor price as

a

and thermal sensor price as

t

, we form two equations:

5a + 3t = 1160

and

3a + 5t = 1080

. Adding both equations gives

8a + 8t = 2240

. Dividing the entire equation by

8

isolates

a + t = 280

Step-by-Step Solution

Define variables and write the system of equations based on the problem statement.

Let

a

be the price of one acoustic sensor and

t

be the price of one thermal sensor.
Equation 1:

5a + 3t = 1160

Equation 2:

3a + 5t = 1080

Translate word problem conditions into algebraic equations.

Add the two equations together to find a direct linear combination.

(5a + 3t) + (3a + 5t) = 1160 + 1080 \implies 8a + 8t = 2240

Since the target is

a + t

, adding symmetric coefficients yields a common factor of

8

, avoiding the need to solve for

a

and

t

individually.

Factor out

8

and solve for

(a + t)

8(a + t) = 2240 \implies a + t = \frac{2240}{8} = 280

Dividing the combined equation by

8

directly isolates the required quantity

(a + t)

Key Concept

Solving Linear Systems via Algebraic Combinations

Question

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Step-by-Step Solution

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