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:

1. \(\rm\frac{p}{x}+\frac{q}{y}\) = m
2. \(\rm\frac{q}{x}+\frac{p}{y}\) = 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:

1. pu + qv = m
2. 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 \(q\):

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

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 \(p\):

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

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 \(\rm\frac{n p−m q}{m p−n q}\).

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.