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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

There are 816 students in enrolled at a certain high school. Each of these students is taking at least one of the subjects economics, geography, and biology. The sum of the number of students taking exactly one of these subjects and the number of students taking all 3 of these subjects is 5 times the number of students taking exactly 2 of these subjects. The ratio of the number of students taking only the two subjects economics and geography to the number of students taking only the two subjects economics and biology to the number of students taking only the two subjects geography and biology is \(3: 6: 8 .\) How many of the students enrolled at this high school are taking only the two subjects geography and biology? a. 35 b. 42 c. 64 d. 136 e. 240

One letter is selected at random from the 5 letters \(\mathrm{V}, \mathrm{W}, \mathrm{X}, \mathrm{Y},\) and \(\mathrm{Z}\), and event \(A\) is the event that the letter \(\mathrm{V}\) is selected. A fair six-sided die with sides numbered \(1,2,3,4,5,\) and 6 is to be rolled, and event \(B\) is the event that a 5 or a 6 shows. A fair coin is to be tossed, and event \(C\) is the event that a head shows. What is the probability that event \(A\) occurs and at least one of the events \(B\) and \(C\) occurs? a. \(\frac{1}{30}\) b. \(\frac{1}{15}\) c. \(\frac{1}{10}\) d. \(\frac{2}{15}\) e. \(\frac{1}{5}\)

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. \(\frac{a c+2 b c}{4}\)

If \(a\) and \(b\) are integers, and \(2 a+b=17,\) then \(8 a+b\) cannot equal which of the following? a. -1 b. 33 c. 35 d. 65 e. 71

Two workers have different pay scales. Worker A receives \(\$ 50\) for any day worked plus \(\$ 15\) per hour. Worker \(B\) receives \(\$ 27\) per hour. Both workers may work for a fraction of an hour and be paid in proportion to their respective hourly rates. If worker A arrives at 9: 21 a.m. and receives the \(\$ 50\) upon arrival and Worker \(\mathrm{B}\) arrives at 10: 09 a.m. assuming both work continuously, at what time would their earnings be identical? a. \(11: 09 \mathrm{a.m}.\) b. \(12: 01 \mathrm{p.m}.\) c. \(2: 31 \mathrm{p.m}.\) d. \(3: 19 \mathrm{p.m}.\) e. \(3: 36 \mathrm{p.m}.\)

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