An art gallery placed two separate orders for exhibition lighting fixtures from the same s… — GMAT

An art gallery placed two separate orders for exhibition lighting fixtures from the same supplier. The first order consisted of $5$ LED spotlights and $3$ halogen floodlights for a total cost of $\$840$ . The second order consisted of $3$ LED spotlights and $5$ halogen floodlights for a total cost of $\$760$ . What is the cost of $1$ LED spotlight?

A
$\$80$
B
$\$100$
$\$120$ Answer
D
$\$150$
E
$\$200$

Answer

The cost of 1 LED spotlight is $120.

Setting up the system of equations

5x + 3y = 840

and

3x + 5y = 760

, adding both equations yields

8x + 8y = 1600

, which simplifies to

x + y = 200

. Subtracting the second equation from the first yields

2x - 2y = 80

, which simplifies to

x - y = 40

. Adding

x + y = 200

and

x - y = 40

gives

2x = 240

, so

x = 120

. Thus,

1

LED spotlight costs

\$120

Step-by-Step Solution

Define variables and set up the system of linear equations based on the two orders.

Let

x

be the cost of

1

LED spotlight and

y

be the cost of

1

halogen floodlight.
Equation (1):

5x + 3y = 840

Equation (2):

3x + 5y = 760

Translating the verbal statements into algebraic equations.

Add Equation (1) and Equation (2) to find the combined linear expression

(x + y)

(5x + 3y) + (3x + 5y) = 840 + 760 \implies 8x + 8y = 1600 \implies x + y = 200

Simplifying the system using linear combination.

Subtract Equation (2) from Equation (1) to find

(x - y)

(5x + 3y) - (3x + 5y) = 840 - 760 \implies 2x - 2y = 80 \implies x - y = 40

Determining the difference between the costs of the two fixtures.

Solve the reduced two-variable system for

x

Add

(x + y = 200)

and

(x - y = 40)

2x = 240 \implies x = 120

Eliminating

y

to determine the exact unit price of

1

LED spotlight.

Key Concept

Solving Systems of Linear Equations via Linear Combination

Estimated Time:1m 30s

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