<PRE><B>Question 201837</B>
let B = weight of  1 barbell
let D = weight of 1 doughnut
let S = weight of 1 sphere
formulas to work with are:
1B + 2S = 24
1B + 2S + 2D = 34
2B + 1S + 1D = 32
first 2 formulas will allow us to get the value of D.
since 1B + 2S = 34 and 1B + 2S + 2D = 35, then
D must = 5
we found this out by subtracting the second equation from the first.
We were left with 2D = 10.
we divided both sides of the equation by 2 to get 
D = 5
now that we know the value of D, we can use that in the remaining equations.
the equation
2B + 1S + 1D = 32 becomes
2B + 1S + 5 = 32
we subtract 5 from both sides of the equation to get
2B + 1S = 27
we can combine this with the first equation of
1B + 2S = 24
to find the value of the remaining variables.
we take the 2 equations and solve them simultaneously.
we have
1B + 2S = 24
2B + 1S = 27
to eliminate one of the variable we need to manipulate one or both of the equations.
we take the first equation and multiply both sides by 2.  This does not change the equality of that equation even though the number of variables involved is different.
that equation becomes
2B + 4S = 48
we now have 2 equations that we subtract from each other to eliminate one of the variables.
those equations are:
2B + 4S = 48
2B + 1S = 27
subtract the bottom one from the top to get
3S = 21
divide both sides of that equation by 3 to get
S = 7
we now know that
D = 5
S = 7
we substitute in any one of the equations to get the remaining variable.
take
2B + 1S = 27
substitute 7 for S to get
2B + 7 = 27
subtract 7 from both sides of the equation to get
2B = 20
divide both sides of the equation by 10 to get
B = 10
we now have:
B = 10
D = 5
S = 7
which is the answer we are looking for.
substituting in all of the equations makes them all true so the values we calculated are good.
for example:
1B + 2S + 2D = 34
substituting in the equations gets
1*10 + 2*7 + 2*5 = 34
which becomes
10 + 14 + 10 = 34
which becomes
34 = 34
which is true.
the other equations check out as well.
your answer is
weight of one barbell = 10 pounds
weight of one sphere = 7 pounds
weight of one doubhnut = 5 pounds
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