For all positive real numbers x and y, if x^2 y^3 = 108 and x^3 y^2 = 72, then x + y = 5. — GMAT

For all positive real numbers $x$ and $y$ , if $x^2 y^3 = 108$ and $x^3 y^2 = 72$ , then $x + y = 5$ .

Cevap: Cevap

The statement is True.

Multiplying the two equations yields

(xy)^5 = 7776 = 6^5

, so

xy = 6

. Dividing the second equation by the first yields

\frac{x}{y} = \frac{2}{3}

, meaning

x = \frac{2}{3}y

. Substituting this into

xy = 6

gives

y = 3

and

x = 2

, making

x + y = 5

Multiply the two system equations to find the product

xy

(x^2 y^3)(x^3 y^2) = x^5 y^5 = (xy)^5 = 108 \times 72 = 7776 = 6^5

, so

xy = 6

Multiplying exponential expressions with common bases allows adding their exponents to form a unified power

(xy)^5

Divide the second equation by the first equation to find the relationship between

x

and

y

\frac{x^3 y^2}{x^2 y^3} = \frac{x}{y} = \frac{72}{108} = \frac{2}{3}, which simplifies to

x = \frac{2}{3}y

Dividing exponential expressions subtracts their exponents, yielding the simple ratio of the variables.

Solve for individual values of

x

and

y

and evaluate

x + y

Substituting

x = \frac{2}{3}y

into

xy = 6

gives

\frac{2}{3}y^2 = 6 \implies y = 3

and

x = 2

. Therefore,

x + y = 5

Solving the system confirms that

x = 2

and

y = 3

are the unique positive real solutions.

System of exponential equations solved by combining power products and ratios.

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