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Zorluk: OrtaSystems of Linear Equations

A manufacturing plant uses two types of machines, Machine X and Machine Y, to produce a certain component. Machine X produces 1515 components per hour, and Machine Y produces 2525 components per hour. On a certain day, both machines were operated for a combined total of 1414 hours, producing a total of 270270 components. For how many hours was Machine X operated?

Cevap: 8 hours

Cevap

Machine X was operated for 8 hours.
By setting up the system of linear equations where xx represents the hours Machine X operated and yy represents the hours Machine Y operated, we get x+y=14x + y = 14 and 15x+25y=27015x + 25y = 270. Substituting y=14−xy = 14 - x into the second equation yields 15x+25(14−x)=27015x + 25(14 - x) = 270. Expanding and simplifying results in −10x+350=270-10x + 350 = 270, which solves to x=8x = 8. Thus, Machine X operated for 8 hours.

Adım Adım Çözüm

1
Define variables for the hours each machine operated and set up the system of linear equations.
Let xx be the hours Machine X was operated and yy be the hours Machine Y was operated. The system is:
x+y=14x + y = 14
15x+25y=27015x + 25y = 270
This translates the verbal descriptions in the problem into algebraic expressions and equations.
2
Express yy in terms of xx from the first equation and substitute it into the second equation.
y=14−xy = 14 - x
15x+25(14−x)=27015x + 25(14 - x) = 270
Substitution reduces the system from two variables to a single equation in terms of xx.
3
Solve the linear equation for xx by distributing the constant and combining like terms.
15x+350−25x=27015x + 350 - 25x = 270
−10x+350=270-10x + 350 = 270
−10x=−80-10x = -80
x=8x = 8
Isolating xx gives the specific number of hours Machine X was operated.

Anahtar Kavram

Solving systems of linear equations in two variables using substitution or elimination.

Alternatif Yöntem

We can solve the system by multiplying the first equation by 25 to get 25x+25y=35025x + 25y = 350. Subtracting the second equation 15x+25y=27015x + 25y = 270 from this yields 10x=8010x = 80, which directly gives x=8x = 8.
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Konular

Systems of Linear Equations

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