Question

If \(x>0, y>0,\) and \(\frac{7 x^{2}+72 x y+4 y^{2}}{4 x^{2}+12 x y+5 y^{2}}=4,\) what is the value of \(\frac{x+y}{y} ?\) a. \(\frac{3}{4}\) b. \(\frac{4}{3}\) c. \(\frac{10}{7}\) d. \(\frac{7}{4}\) e. \(\frac{7}{3}\)

Short answer

\( \frac{7}{3} \)

Step-by-step solution

Simplify the given equation

Start with the provided equation: \[ \frac{7 x^{2}+72 x y+4 y^{2}}{4 x^{2}+12 x y+5 y^{2}}=4 \]Multiply both sides by the denominator to eliminate the fraction: \[ 7 x^2 + 72 xy + 4 y^2 = 4 (4 x^2 + 12 xy + 5 y^2) \]

Expand and simplify

Expand the right-hand side of the equation: \[ 7 x^2 + 72 xy + 4 y^2 = 16 x^2 + 48 xy + 20 y^2 \]Move all terms to one side to set the equation to zero: \[ 7 x^2 + 72 xy + 4 y^2 - 16 x^2 - 48 xy - 20 y^2 = 0 \]Combine like terms: \[ -9 x^2 + 24 xy - 16 y^2 = 0 \]

Factorize the equation

Notice that we can factor the quadratic equation: \[ - 9 x^2 + 24 xy - 16 y^2 = (3x - 4y)(3x - 4y) = 0 \]This implies: \[ 3x - 4y = 0 \]

Solve for the ratio \( \frac{x}{y} \)

Since \( 3x - 4y = 0 \), it simplifies to: \[ 3x = 4y \]So, \( \frac{x}{y} = \frac{4}{3} \)

Calculate \( \frac{x + y}{y} \)

By substituting \( \frac{x}{y} = \frac{4}{3} \), we get: \[ \frac{x + y}{y} = \frac{x}{y} + 1 = \frac{4}{3} + 1 = \frac{4}{3} + \frac{3}{3} = \frac{7}{3} \]

Key concepts

Rational Equations

Rational equations are equations that involve fractions containing algebraic expressions in the numerator and denominator. To solve these, the first step is often to eliminate the fraction by multiplying both sides by the denominator. In the problem, the given equation is: \[ \frac{7x^2 + 72xy + 4y^2}{4x^2 + 12xy + 5y^2} = 4 \] We eliminate the fraction by multiplying both sides by \( 4x^2 + 12xy + 5y^2 \). This results in a new equation without fractions: \[ 7x^2 + 72xy + 4y^2 = 4(4x^2 + 12xy + 5y^2) \] Clearing fractions is a critical first step in solving rational equations, making them easier to manipulate and solve.

Factoring Quadratics

Factoring quadratics is a method used to simplify polynomial equations, especially useful for solving quadratic equations, like in our problem. After expanding and rearranging the equation, we get: \[ -9x^2 + 24xy - 16y^2 = 0 \] We notice that this quadratic can be factored as a perfect square trinomial: \[ (3x - 4y)^2 = 0 \] Factoring simplifies the complex polynomial into a product of simpler binomials. Since \( (3x - 4y)^2 = 0 \), it directly provides us with: \[ 3x - 4y = 0 \] Factoring helps break down seemingly complicated expressions into simpler, manageable ones, facilitating easier solutions.

Algebraic Manipulation

Algebraic manipulation involves using mathematical operations to rearrange and simplify equations. After factoring, we solve for one variable in terms of another. From: \[ 3x - 4y = 0 \] Solving for \( x \) in terms of \( y \): \[ 3x = 4y \] \[ \frac{x}{y} = \frac{4}{3} \] Finally, we use this relationship to find \( \frac{x + y}{y} \): \[ \frac{x+y}{y} = \frac{x}{y} + 1 = \frac{4}{3} + 1 = \frac{4}{3} + \frac{3}{3} = \frac{7}{3} \] Algebraic manipulation allows us to express complex relationships in simpler forms and solve for the desired quantities.