SOLUTION: Bonnie has three types of weights. Barbells all have the same weight. Doughnuts all have the same weight. Spheres all have the same weight.. She does not know haw much each eaight

Question 201837: Bonnie has three types of weights. Barbells all have the same weight. Doughnuts all have the same weight. Spheres all have the same weight.. She does not know haw much each eaight weighs but she does know, a barbell and two spheres weigh 24lbs, a barbell, 2 doughnuts and 2 spheres weigh 34 lbs, and 2 barbells, 1 doughnut and 1 sphere weigh 32 lbs. How much do the individuals weigh?
Found 2 solutions by solver91311, Theo:
Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!

Let represent the weight of one barbell.

Let represent the weight of one doughnut.

Let represent the weight of one sphere.

A barbell and two spheres weigh 24 lbs:

Two doughnuts and two spheres weigh 34 lbs:

Two barbells, one doughnut, and one sphere weigh 32 lbs:

Solve the system for the ordered triple:

John

Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
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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