If a, b, and c are positive real numbers such that a(b + c) = 20, b(c + a) = 13, and c(a +… — ACT

If $a$ , $b$ , and $c$ are positive real numbers such that $a(b + c) = 20$ , $b(c + a) = 13$ , and $c(a + b) = 25$ , what is the value of the product $abc$ ?

Answer: 24

Answer

The product

abc

is equal to 24.

By applying the distributive property, we expand the system to

ab + ac = 20

ab + bc = 13

, and

bc + ac = 25

. Adding these yields

2(ab + bc + ac) = 58

, so

ab + bc + ac = 29

. We isolate the individual products:

bc = 9

ac = 16

, and

ab = 4

. Multiplying these gives

(abc)^2 = 4 \times 9 \times 16 = 576

. Since the variables are positive,

abc = \sqrt{576} = 24

Step-by-Step Solution

Expand the equations using the distributive property.

ab + ac = 20

ab + bc = 13

, and

bc + ac = 25

This allows us to work with the pairwise products

ab

bc

, and

ac

directly.

Sum the three equations and divide by 2.

ab + bc + ac = 29

By adding the equations, each pairwise product appears twice:

(ab + ac) + (ab + bc) + (bc + ac) = 2(ab + bc + ac) = 58

. Dividing by 2 gives their sum.

Solve for each pairwise product by subtracting the original equations from the sum.

bc = 9

ac = 16

, and

ab = 4

Subtracting

ab + ac = 20

from

ab + bc + ac = 29

isolates

bc = 9

. Similarly, subtracting

ab + bc = 13

isolates

ac = 16

, and subtracting

bc + ac = 25

isolates

ab = 4

Multiply the pairwise products and take the square root.

(abc)^2 = 576 \implies abc = 24

Multiplying

(ab)(bc)(ac)

gives

(abc)^2

. Since

a

b

, and

c

are positive, their product

abc

must also be positive, so we take the positive square root of

576

, which is

24

Key Concept

Distributive property of multiplication over addition, and properties of equality in systems of equations.

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Topics

More questions from this topic