A bicycle manufacturer builds racing bikes and mountain bikes. Materials for the racing bike cost $110 while labor to build them is $120. Materials for the mountain bike cost $140 and labor is $180. The company budgeted $31,800 for labor and $26,150 for materials. How many of each bicycle did they build?
linear programming
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nr = number of racing bikes
nm = number of mountain bikes
lc = labor cost
lc = 120 nr + 180 nm
lc </= $31,800
mc = materials cost
mc = 110 nr + 140 nm
mc /= $26,150
sorry, do not need programming, nothing to maximize oer minimize, simple solution of two equations, two unknowns
120 nr + 180 nm = $31,800
110 nr + 140 nm = $26,150
solve for nr and nm
Ya but how would you solve for it using Substitution or Elimination?
To solve this problem, we can use a system of linear equations. Let's define the variables:
Let R be the number of racing bikes.
Let M be the number of mountain bikes.
Now we can set up the equations based on the given information:
1. Total Labor Cost Equation: 120R + 180M = 31,800
The labor cost per racing bike is $120, so the total labor cost for all the racing bikes is 120R. Similarly, the total labor cost for all the mountain bikes is 180M. The sum of these two costs should equal the total labor budget of $31,800.
2. Total Material Cost Equation: 110R + 140M = 26,150
The material cost per racing bike is $110, so the total material cost for all the racing bikes is 110R. Similarly, the total material cost for all the mountain bikes is 140M. The sum of these two costs should equal the total material budget of $26,150.
Now we have a system of two equations:
120R + 180M = 31,800
110R + 140M = 26,150
To solve this system, we can use various methods such as substitution or elimination. Here, let's use the elimination method.
Multiply the first equation by 11 and the second equation by 12 to eliminate the R term:
1,320R + 1,980M = 349,800
1,320R + 1,680M = 313,800
Now subtract the second equation from the first:
1,320R + 1,980M - 1,320R - 1,680M = 349,800 - 313,800
Simplifying the equation:
300M = 36,000
Dividing both sides of the equation by 300:
M = 120
Substitute the value of M back into one of the original equations. Let's use the first equation:
120R + 180(120) = 31,800
Simplifying the equation:
120R + 21,600 = 31,800
Subtracting 21,600 from both sides of the equation:
120R = 10,200
Dividing both sides of the equation by 120:
R = 85
Therefore, the company built 85 racing bikes and 120 mountain bikes.