How to Formulate an Equation Representing a Problem

First, write down the sentences you can turn into equations. Then, translate them into equations:

Einstein is $101$ years older than Cooper.

$$E=101+C$$

Hawking is $38$ years older than Cooper.

$$H=38+C$$

Added together, they’re $250$ years old.

$$E+C+H=250$$

To solve this, you can just insert the expressions for $E$ and $H$ into the third equation:

That means Sheldon Cooper is $37$ years old, which makes Albert Einstein $37+101=138$ years old and makes Stephen Hawking $37+38=75$ years old.

The first thing you have to do with word problems is to write each sentence as an equation. Look at this:

Item 1:

Four adult tickets and two children’s tickets cost $$440$. If you know the price of an adult ticket $x$ and the price of a children’s ticket for $y$, you can translate this to:

$$4x+2y=440.$$

Item 2:

One adult ticket and four children’s tickets cost $$320$. You can translate this into:

$$x+4y=320.$$

Then you have two equations with two variables. That means you can solve them by either using the substitution method, or the elimination method. Here, I’ve used the substitution method, because that’s my favorite.

Substituting (2) into (1) gives:

$$\begin{array}{llll}\hfill 4(320-4y)+2y& =440& \hfill & \\ \hfill 1280-16y+2y& =440& \hfill & \\ \hfill -14y& =440-1280& \hfill & \\ \hfill \frac{-14y}{-14}& =\frac{-840}{-14}& \hfill & \\ \hfill y& =60& \hfill & \end{array}$$

Inserting this value back into (2):

$$\begin{array}{llll}\hfill x& =320-4\cdot 60& \hfill & \\ \hfill x& =320-240& \hfill & \\ \hfill x& =80& \hfill & \end{array}$$

The price of an adult ticket is $$80$, and the price of a children’s ticket is $$60$.