SOLUTION: I have no idea how to solve this problem. Your help would be appreciated...thanx!!! Identify two variables (x and y), write a system of equations and solve by either the additi

Algebra ->  Coordinate Systems and Linear Equations -> SOLUTION: I have no idea how to solve this problem. Your help would be appreciated...thanx!!! Identify two variables (x and y), write a system of equations and solve by either the additi      Log On


   

Question 141: I have no idea how to solve this problem. Your help would be appreciated...thanx!!!
Identify two variables (x and y), write a system of equations and solve by either the addition/elimination method or substitution method. Respond in a complete sentence.
Word Problem: The readings from two different pollution gauges are recorded at the end of each shift in a factory. These readings are compared to see which reading is higher and totaled to get an overview of factorywide pollution. On one reading the first gauge totaled 7.5 units more than twice the reading on the other gauge. If the total of these readings was 42 units, what was the reading on each gauge?
Answer by KMST(5473) About Me  (Show Source):
You can put this solution on YOUR website!
Let x be the first gauge reading, and y be the reading on the other gauge.
"On one reading the first gauge totaled 7.5 units more than twice the reading on the other gauge" translates into
x=2y%2B7.5 ,
and "the total of these readings was 42 units" translates into
x%2By=42 .
The system of equations to solve is system%28x=2y%2B7.5%2Cx%2By=42%29

SOLVING BY SUBSTITUTION works very well in this case and may be a good option in some other cases:
FIRST, we find an expression that gives us the value of one variable calculated from just the value of the other.
In some cases that would mean solve an equation for x of for y.
No need in this case.
The top equation tells us that the value of x can be calculated as 2y%2B7.5 (equation 1).
NEXT, we SUBSTITUTE that expression for the one variable into the second equation:
x%2By=42 (equation 2) --> %282y%2B7.5%29%2By=42 , and solve for the second variable
%282y%2B7.5%29%2By=42--> 2y%2B7.5%2By=42 --> 3y%2B7.5=42 --> 3y=42-7.5 --> 3y=34.5 --> y=34.5%2F3 --> y=highlight%2811.5%29
FINALLY, we SUBSTITUTE the value found for the second variable into the first equation:
x=2y%2B7.5 --> x=2%2A11.5%2B7.5 --> x=23%2B7.5 --> highlight%28x=30.5%29

One sentence solution:
The reading on the first gauge was 30.5 units and the reading on the other gauge was 11.5 units.

SOLVING BY ELIMINATION works better in some cases:
FIRST, we make a combination of the two equations in the system in such a way that we ELIMINATE one variable.
In our case, we have
equation 1 x=2y%2B7.5 , and
equation 2 x%2By=42 .
To combine equations, we can multiply each equation by a number and add up the resulting equations.
Lets say we multiply equation 2 times 1 ,
and add it to equation 1 multiplied times %28-1%29 , -x=-2y-7.5 ,

NEXT we solve the resulting equation
y=-2y%2B34.5 --> 3y=34.5 --> y=34.5%2F3 --> y=highlight%2811.5%29
FINALLY, we SUBSTITUTE the value found for that variable in one of the equations ,
and solve to find the value of the other variable:
x=2y%2B7.5 --> x=2%2A11.5%2B7.5 --> x=23%2B7.5 --> highlight%28x=30.5%29 .