For the equation $py = xy + 1$ where $p$ is an unknown constant, may be represented by a straight line, expressed in the form $Y = mX + c$, where $X$ and $Y$ are functions of $x$ and/or $y$ and $m$ and $c$ are constants. Using the following table, insert in it an expression for $Y$, $X$, $m$ and $c.$ The variables $x$ and $y$ are related by the equation $x^{a}y = b$, where $a$ and $b$ are constants. The table below shows some corresponding values of $x$ and $y$. Graph paper for plotting lg y against lg x is provided on this page. Plot lg y against lg x and draw a straight line graph. Use your graph to estimate the value of $a$ and of $b$.

1. Rearrange the given equation to isolate a term involving only $y$ or to separate the variables into a linear form $Y = mX + c$.

Given $py = xy + 1$.
Divide the entire equation by $y$:
$p = x + \frac{1}{y}$

2. Express the equation in the form $Y = mX + c$ and identify the components.

Rearranging the terms gives:
$\frac{1}{y} = -x + p$
Comparing this with $Y = mX + c$, we can identify:
$Y = \frac{1}{y}$
$X = x$
$m = -1$
$c = p$

1. Calculate the values of $\lg x$ and $\lg y$ to 3 significant figures.

Using the given table:

2. Plot the points and draw the straight line graph.

Plot the points $(0.117, 3.104)$, $(0.212, 2.600)$, $(0.294, 2.117)$, $(0.405, 1.545)$, and $(0.605, 0.420)$ on the graph paper with $\lg y$ on the vertical axis and $\lg x$ on the horizontal axis. Draw a straight line of best fit passing through or near these points.

1. Convert the non-linear equation to linear form

Given $x^a y = b$.
Taking logarithms on both sides:
$\lg(x^a y) = \lg b$
$a \lg x + \lg y = \lg b$
$\lg y = -a \lg x + \lg b$
This is in the form $Y = mX + c$, where $Y = \lg y$, $X = \lg x$, gradient $m = -a$, and $Y$-intercept $c = \lg b$.

2. Calculate the gradient to find $a$

Using the points $(\lg 1.31, \lg 1270)$ and $(\lg 4.03, \lg 2.63)$, which are approximately $(0.117, 3.104)$ and $(0.605, 0.420)$:
Gradient $m = \frac{0.420 - 3.104}{0.605 - 0.117} = \frac{-2.684}{0.488} \approx -5.5$
Since $m = -a$, we have $-a = -5.5 \implies a = 5.5$.

3. Find the $Y$-intercept to determine $b$

Using the point $(0.117, 3.104)$ and $m = -5.5$:
$c = Y - mX = 3.104 - (-5.5)(0.117) = 3.104 + 0.6435 \approx 3.75$
Since $c = \lg b$, we have $\lg b = 3.75$.

4. Calculate the value of $b$

$b = 10^{3.75} \approx 5620$ (accept values around 5600 depending on graph reading)