Simple Equations - Setting up an equation

An equation is a mathematical statement of equality containing one or more variables. It can be visualized as a balanced weighing scale where the '=' sign acts as the fulcrum (pivot). For the scale to stay level, the total value on the Left Hand Side (LHS) must be exactly equal to the total value on the Right Hand Side (RHS).

Variables are unknown values represented by letters like $x, y, z, l, m,$ or $n$. Think of a variable as an 'empty box' whose contents we need to find. In contrast, constants are fixed numerical values like $5, -10,$ or $\frac{1}{2}$ that do not change.

Setting up an equation involves translating verbal phrases into mathematical operations. Phrases like 'sum of' or 'added to' imply addition ($+$), 'difference between' or 'subtracted from' imply subtraction ($-$), 'times' or 'product' imply multiplication ($\times$), and 'divided by' implies division ($\div$).

The phrase 'is' or 'gives' or 'results in' represents the equals sign ($=$) in an equation. For example, 'The sum of $x$ and $5$ is $12$' translates to $x + 5 = 12$. Visually, this means a block of length $x$ joined with a block of length $5$ matches a single block of length $12$ exactly.

In word problems, identifying the unknown quantity is the first step. We assign a variable to this unknown. For instance, if a problem asks for 'the number of marbles Rahul has,' we let that number be $m$. Every other piece of information in the problem is then written in relation to $m$.

Equations can involve multiple operations. For example, '3 more than twice a number is 11' involves both multiplication and addition. You first multiply the variable by 2 ($2x$) and then add 3 ($2x + 3$). The final equation is $2x + 3 = 11$.

An equation remains balanced if you perform the same operation on both sides. Imagine adding an equal weight to both pans of a scale; the scale remains balanced. Similarly, if $x + 2 = 5$, then $(x + 2) - 2 = 5 - 2$ also holds true.