The equation y = p times the square root of x plus q divided by the square root of x is non-linear. Change it to a linear form.

Step-by-step solution

1. 1

   Fix what "linear form" is asking for. The goal is an equation $Y=mX+c$ in which $Y$ and $X$ are expressions built from $x$ and $y$, and the unknown constants $p$ and $q$ appear only as the gradient and the intercept. Then plotting $Y$ against $X$ from experimental data gives $p$ and $q$ from the line.

2. 2

   Identify what blocks that form. As written, the right-hand side contains both $\sqrt{x}$ and $x^{-1/2}$. No choice of $X$ makes a sum of two different powers linear, so the equation has to be reshaped before any variable is renamed.

3. 3

   Multiply both sides by $\sqrt{x}$. Since $x>0$ is required for $\sqrt{x}$ to be defined and nonzero, this is a legal operation:

   $y\sqrt{x}=p\sqrt{x}\cdot\sqrt{x}+\frac{q}{\sqrt{x}}\cdot\sqrt{x}=px+q$

   The two awkward half-powers collapse: one becomes $x$, the other becomes a constant.

4. 4

   Name the new variables and read off the constants. Setting

   $Y=y\sqrt{x},\qquad X=x$

   the relation is $Y=pX+q$: a straight line whose gradient is $p$ and whose vertical intercept is $q$.

5. 5

   Check with concrete numbers. Take $p=2$, $q=3$ and $x=4$. The original equation gives $y=2(2)+\tfrac{3}{2}=5.5$, so $Y=y\sqrt{x}=5.5\times 2=11$. The linear form predicts $pX+q=2(4)+3=11$ — identical, confirming the transformation preserves the relation rather than just looking tidier.