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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

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

Step by step solution

01

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: \[ 7 x^2 + 72 xy + 4 y^2 = 4 (4 x^2 + 12 xy + 5 y^2) \]
02

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: \[ -9 x^2 + 24 xy - 16 y^2 = 0 \]
03

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: \[ 3x - 4y = 0 \]
04

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

Since \( 3x - 4y = 0 \), it simplifies to: \[ 3x = 4y \]So, \( \frac{x}{y} = \frac{4}{3} \)
05

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

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

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

A car traveled from Town \(A\) to Town \(B\). The car traveled the first \(\frac{3}{8}\) of the distance from Town \(A\) to Town \(B\) at an average speed of \(x\) miles per hour, where \(x>0 .\) The car traveled the remaining distance at an average speed of \(y\) miles per hour, where \(y>0 .\) The car traveled the entire distance from Town \(A\) to Town \(B\) at an average speed of \(z\) miles per hour. Which of the following equations gives \(y\) in terms of \(x\) and \(z ?\) a. \(y=\frac{3 x+5 z}{8}\) b. \(y=\frac{5 x z}{8 x+3 z}\) c. \(y=\frac{8 x-3 z}{5 x z}\) d. \(y=\frac{3 x z}{8 x-5 z}\) e. \(y=\frac{5 x z}{8 x-3 z}\)

A department of motor vehicles asks visitors to draw numbered tickets from a dispenser so that they can be served in order by number. Six friends have graduated from truck-driving school and go to the department to get commercial driving licenses. They draw tickets and find that their numbers are a set of evenly spaced integers with a range of 10 Which of the following could NOT be the sum of their numbers? a. 1,254 b. 1,428 c. 3,972 d. 4,316 e. 8,010

Set \(X\) consists of at least 2 members and is a set of consecutive odd integers with an average (arithmetic mean) of 37. Set \(Y\) consists of at least 10 members and is also a set of consecutive odd integers with an average (arithmetic mean) of 37. Set \(Z\) consists of all of the members of both set \(X\) and set \(Y\). Which of the following statements must be true? I. The standard deviation of set \(Z\) is not equal to the standard deviation of set \(X\). II. The standard deviation of set \(Z\) is equal to the standard deviation of set \(Y\). III. The average (arithmetic mean) of set \(Z\) is 37. a. I only b. II only c. III only d. I and III e. II and III

When the cube of a non-zero number \(y\) is subtracted from \(35,\) the result is equal to the result of dividing 216 by the cube of that number \(y .\) What is the sum of all the possible values of \(y ?\) a. \(\frac{5}{2}\) b. 5 c. 6 d. 10 e. 12

Working alone at a constant rate, machine \(\mathrm{P}\) produces \(a\) widgets in 3 hours. Working alone at a constant rate, machine \(\mathrm{Q}\) produces \(b\) widgets in 4 hours. If machines \(P\) and \(Q\) work together for \(c\) hours, then in terms of \(a, b,\) and \(c,\) how many widgets will machines \(\mathrm{P}\) and \(\mathrm{Q}\) produce? a. \(\frac{3 a c+4 b c}{12}\) b. \(\frac{4 a c+3 b c}{12}\) c. \(\frac{4 a c+3 b c}{6}\) d. \(4 a c+3 b c\) e. \(\frac{a c+2 b c}{4}\)

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