How to solve systems of linear equations with three unknowns

Question

A manufacturer produces 40-inch, 46-inch and 52-inch television sets that require assembly, testing and packing. Each 40-inch set requires 45 minutes to assemble, 30 minutes to test and 10 minutes to package. Each 46-inch set requires 1 hour to assemble, 45 minutes to test and 15 minutes to package, Each 52-inch set requires 1.5 hours to assemble, 1 hour to test and 15 minutes to package. If the assembly line operates for 17.75 hours per day, the test facility is used for 12.5 hours per day and the packing equipment is used for 3.75 hours per day, how many of each type of set can be produced?

Answer 1

The number of unknowns tells you how many equations you mush have. For this problem, I recommend one equation for assembly, one for testing, and one for packaging. First, decide which variables you'll use, and then define what they represent. (I chose a=# of 40" tvs, b=# of 46" tvs, c=# of 52" tvs). Since assembly requires 0.75 hrs for the 40", 1 hr for the 46", and 1.5 hrs for the 52", the assembly equation should be .75a+1b+1.5c=17.75. The other 2 equations will be similar.

Once you have all 3 equations, you must go thru the process of solving for each variable. You can use matrices if you're familiar with the process. But if not, you start by eliminating a variable from 1 PAIR of equations, then eliminate the SAME variable from a DIFFERENT pair of equations. Then you can use your 2 new, 2 variable equations to solve for one variable, using either elimination or substitution. Then just go in reverse... take the value you got on the last step, and substitute it back in to one of your 2 variable equations to find the value of another variable. Then take both values and substitute them back in to one of your original equations to find the 3rd value.

Be sure to check your solution by substituting ALL 3 values into ALL 3 original equations to see if they work. If they do, your solution is correct.

Please let me know if you need further assistance.

Answer 2