# Real-life equations

Here are several equations from real life. Can you work out which measurements are possible from each equation?

This is a list of many of the most important equations in science. In each case, we have labelled the two variable quantities $x$ and $y$. The letters $a, b$ stand for constants in each case

Constant motion $\quad\quad\quad\quad\quad a = \frac{x}{y}$

Constant acceleration $\quad\quad\quad x = uy + \frac{1}{2} ay^2$

Beer Lambert Law $\quad\quad\quad\quad a=bxy$

Exponential decay $\quad\quad\quad\quad x=a e^{by}$

Michaelis-Menton $\quad\quad\quad\quad x = \frac{ay}{b+y}$

pH $\quad\quad\quad\quad\quad\quad\quad\quad\quad x = -\log_{10}(y)$

Can you identify the possible meanings of the variables $x$ and $y$ and the constants in each case?

Image

Four graphs are shown above, where the two axes intersect at the origin $(0, 0)$.

The red crosses show four measurements. Although we do not know the numerical values (because there are no scales on the graphs), we can see whether the values are positive or negative in each variable. For example, the first measurement is positive in $x$ and positive in $y$; the second measurement is positive in $y$, negative in $x$.

For processes evolving according to each of the equations above, which measurements are possible?

### Why do this problem?

This problem encourages students to get into the real meaning of equations and graphical representation without getting bogged down in algebraic calculations or falling back into blind computation. It will help to reinforce the differences between different 'types' of equation.### Possible approach

Note the difference between showing that an equation
is a possibility and
showing that it is not a
possibility. In the first case, students need only give a single
example of a curve with certain paramaters which passes through a
point of the required type. To show that an equation CANNOT pass
through a point of a certain type requires more careful
explanation. Hopefully students will work this out for themselves,
but prompt them if necessary.

### Key questions

- How can you tell if a certain point will match a certain equation type?
- How can you tell if a certain point will not match a certain equation type?

### Possible extension

You might naturally try Equation
matching next.

### Possible support

Give concrete examples by labelling the points $(1, -1), (-1,
1), (-1, -1), (1, -1)$

Alternatively, try the easier non-algebraic question Bio-graphs