- #1

Dukon

- 73

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[itex] { A \over B - C} = {D \over B} + {E \over C } [/itex]

Given the above equation and that A,B,C are known, how are the only two unknowns D and E determined from the knowns: A,B,C?

By the method of partial fractions, I could let [itex] b \equiv \sqrt{B} [/itex] and [itex] c \equiv \sqrt{C} [/itex] to rewrite the target where [itex] B-C = (b+c)(b-c) [/itex] but I see that as a dead end because D is defined over B not over (b+c). Therefore, pursuing a different approach the original equation may be written:

[tex] { A \over B - C } = {A \over B (1-{C\over B}) } = { D(1 - {C\over B}) \over B (1 - {C\over B}) } + { E \left( {B \over C} \left( 1-{C\over B} \right) \right) \over C \left( {B \over C} \left( 1-{C\over B} \right) \right)} [/tex]

At this point all denominators are same so now exists one equation in two unknowns D & E:

[tex] A = { D(1 - {C\over B}) } + { E \left( {B \over C} \left( 1-{C\over B} \right) \right) } = { D(1 - {C\over B}) } + { E \left( {B \over C} - 1 \right) } [/tex]

where the one equation in two unknowns is:

[tex] D = {A \over (1 - {C\over B}) } + E = {A \over B - C} B + E [/tex]

But what can I use as the second equation?

Given the above equation and that A,B,C are known, how are the only two unknowns D and E determined from the knowns: A,B,C?

By the method of partial fractions, I could let [itex] b \equiv \sqrt{B} [/itex] and [itex] c \equiv \sqrt{C} [/itex] to rewrite the target where [itex] B-C = (b+c)(b-c) [/itex] but I see that as a dead end because D is defined over B not over (b+c). Therefore, pursuing a different approach the original equation may be written:

[tex] { A \over B - C } = {A \over B (1-{C\over B}) } = { D(1 - {C\over B}) \over B (1 - {C\over B}) } + { E \left( {B \over C} \left( 1-{C\over B} \right) \right) \over C \left( {B \over C} \left( 1-{C\over B} \right) \right)} [/tex]

At this point all denominators are same so now exists one equation in two unknowns D & E:

[tex] A = { D(1 - {C\over B}) } + { E \left( {B \over C} \left( 1-{C\over B} \right) \right) } = { D(1 - {C\over B}) } + { E \left( {B \over C} - 1 \right) } [/tex]

where the one equation in two unknowns is:

[tex] D = {A \over (1 - {C\over B}) } + E = {A \over B - C} B + E [/tex]

But what can I use as the second equation?

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