If $\rm\frac{p}{x}+\frac{q}{y}$ = m and $\rm\frac{q}{x}+\frac{p}{y}$ = n, then what is $\rm\frac{x}{y}$ equal to?

Question

If $\rm\frac{p}{x}+\frac{q}{y}$  = m and  $\rm\frac{q}{x}+\frac{p}{y}$  = n, then what is  $\rm\frac{x}{y}$ equal to?

Prepp Team · Accepted Answer

Solving Simultaneous Equations to Find a Ratio

The problem asks us to find the value of the ratio $\rm\frac{x}{y}$ given two equations involving $\rm\frac{1}{x}$ and $\rm\frac{1}{y}$. We are given:

$\rm\frac{p}{x}+\frac{q}{y}$ = m
$\rm\frac{q}{x}+\frac{p}{y}$ = n

We can treat this as a system of linear equations in terms of $\rm\frac{1}{x}$ and $\rm\frac{1}{y}$. Let $u = \rm\frac{1}{x}$ and $v = \rm\frac{1}{y}$. The equations become:

pu + qv = m
qu + pv = n

We want to find $\rm\frac{x}{y}$. Since $u = \rm\frac{1}{x}$ and $v = \rm\frac{1}{y}$, we have $x = \rm\frac{1}{u}$ and $y = \rm\frac{1}{v}$. Therefore, $\rm\frac{x}{y} = \frac{1/u}{1/v} = \frac{v}{u}$. So, our goal is to solve for $u$ and $v$ and find their ratio $\frac{v}{u}$.

Solving the System using Elimination

We can use the elimination method to solve for $u$ and $v$.

To eliminate $v$, multiply equation (1) by $p$ and equation (2) by $q$:

$p(pu + qv) = pm \implies p^2 u + pqv = pm$
$q(qu + pv) = qn \implies q^2 u + pqv = qn$

Now, subtract the new equation (2) from the new equation (1):

$(p^2 u + pqv) - (q^2 u + pqv) = pm - qn$

$p^2 u - q^2 u = pm - qn$

$(p^2 - q^2) u = pm - qn$

If $p^2 - q^2 \neq 0$, we can solve for $u$:

$u = \frac{pm - qn}{p^2 - q^2}$

To eliminate $u$, multiply equation (1) by $q$ and equation (2) by $p$:

$q(pu + qv) = qm \implies pqu + q^2v = qm$
$p(qu + pv) = pn \implies pqu + p^2v = pn$

Now, subtract the new equation (1) from the new equation (2):

$(pqu + p^2v) - (pqu + q^2v) = pn - qm$

$p^2 v - q^2 v = pn - qm$

$(p^2 - q^2) v = pn - qm$

If $p^2 - q^2 \neq 0$, we can solve for $v$:

$v = \frac{pn - qm}{p^2 - q^2}$

Finding the Ratio $\rm\frac{v}{u}$ which is $\rm\frac{x}{y}$

Now we have the expressions for $u$ and $v$. We can find the ratio $\frac{v}{u}$:

$\frac{v}{u} = \frac{\frac{pn - qm}{p^2 - q^2}}{\frac{pm - qn}{p^2 - q^2}}$

Assuming $p^2 - q^2 \neq 0$ and $pm - qn \neq 0$, we can cancel the denominator $p^2 - q^2$:

$\frac{v}{u} = \frac{pn - qm}{pm - qn}$

Since $\rm\frac{x}{y} = \frac{v}{u}$, the value of $\rm\frac{x}{y}$ is $\frac{pn - qm}{pm - qn}$.

Let's compare this with the given options. Note that the terms in the numerator and denominator can be written in different orders due to commutativity of multiplication and addition/subtraction (with sign). The expression $\frac{pn - qm}{pm - qn}$ is equivalent to $\frac{np - mq}{mp - nq}$.

Matching with Options

Let's check the options provided:

Option 1: $\rm\frac{n p+m q}{m p+n q}$
Option 2: $\rm\frac{n p+m q}{m p−n q}$
Option 3: $\rm\frac{n p−m q}{m p−n q}$
Option 4: $\rm\frac{n p−m q}{m p+n q}$

Our result $\frac{np - mq}{mp - nq}$ matches Option 3.

The final answer is $\rm\frac{n p−m q}{m p−n q}$.

Equation 1	$\rm\frac{p}{x}+\frac{q}{y}$ = m
Equation 2	$\rm\frac{q}{x}+\frac{p}{y}$ = n
Substitution	$u = \rm\frac{1}{x}$, $v = \rm\frac{1}{y}$
New Equations	pu + qv = m qu + pv = n
Solution for u	$u = \frac{pm - qn}{p^2 - q^2}$
Solution for v	$v = \frac{pn - qm}{p^2 - q^2}$
Ratio $\rm\frac{x}{y} = \frac{v}{u}$	$\frac{pn - qm}{pm - qn}$ or $\frac{np - mq}{mp - nq}$

Revision Table: Key Steps in Solving for $\rm\frac{x}{y}$

Identify the given equations and the required expression $\rm\frac{x}{y}$.
Recognize the structure of the equations as linear in terms of reciprocals ($\rm\frac{1}{x}$ and $\rm\frac{1}{y}$).
Substitute variables (e.g., $u=\rm\frac{1}{x}, v=\rm\frac{1}{y}$) to simplify the system.
Solve the resulting linear system for the new variables ($u$ and $v$) using methods like elimination.
Express the desired ratio $\rm\frac{x}{y}$ in terms of the new variables ($\frac{v}{u}$).
Substitute the solutions for $u$ and $v$ into the ratio expression and simplify.
Match the final simplified expression with the given options.

Additional Information: Systems of Linear Equations

A system of linear equations is a set of two or more linear equations involving the same variables. A solution to a system of linear equations is a set of values for the variables that satisfies all the equations simultaneously.

Common methods for solving systems of linear equations include:

Substitution Method: Solve one equation for one variable and substitute that expression into the other equation.
Elimination Method: Multiply equations by constants so that when the equations are added or subtracted, one variable is eliminated.
Matrix Methods: Represent the system using matrices and use techniques like Gaussian elimination or Cramer's rule (for certain systems).

In this problem, although the original equations involve reciprocals, the system becomes a standard linear system after substituting $u=\rm\frac{1}{x}$ and $v=\rm\frac{1}{y}$. The choice of method (substitution or elimination) depends on personal preference and the specific form of the equations, but elimination was straightforward here.

The existence of a unique solution for $u$ and $v$ (and thus for $x$ and $y$) depends on the coefficients $p$ and $q$ and whether $p^2 - q^2 \neq 0$. The existence of the ratio $\frac{v}{u}$ further assumes $u \neq 0$, which means $pm - qn \neq 0$. If these conditions are not met, the system might have no unique solution, infinite solutions, or the ratio $\frac{x}{y}$ might be undefined or zero.

Prepp Team · Answer

$\rm\frac{n p+m q}{m p+n q}$

Prepp Team · Answer

$\rm\frac{n p+m q}{m p&minus;n q}$

Prepp Team · Answer

$\rm\frac{n p&minus;m q}{m p+n q}$

Solving Simultaneous Equations to Find a Ratio

Solving the System using Elimination

Finding the Ratio \(\rm\frac{v}{u}\) which is \(\rm\frac{x}{y}\)

Matching with Options

Revision Table: Key Steps in Solving for \(\rm\frac{x}{y}\)

Additional Information: Systems of Linear Equations

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Equation 1	\(\rm\frac{p}{x}+\frac{q}{y}\) = m
Equation 2	\(\rm\frac{q}{x}+\frac{p}{y}\) = n
Substitution	\(u = \rm\frac{1}{x}\), \(v = \rm\frac{1}{y}\)
New Equations	pu + qv = m qu + pv = n
Solution for u	\(u = \frac{pm - qn}{p^2 - q^2}\)
Solution for v	\(v = \frac{pn - qm}{p^2 - q^2}\)
Ratio \(\rm\frac{x}{y} = \frac{v}{u}\)	\(\frac{pn - qm}{pm - qn}\) or \(\frac{np - mq}{mp - nq}\)