An equation that contains only one unknown quantity or one quantity that can vary is called an equation in one variable.

For example, the equation $2\ell + 2w = 72$ represents the relationship between the length, $\ell$, and the width, $w$, of a rectangle that has a perimeter of 72 units. If we know that the length is 15 units, we can rewrite the equation as:

$2(15) + 2w = 72$.

This is an equation in one variable, because $w$ is the only quantity that we don't know. To solve this equation means to find a value of $w$ that makes the equation true.

In this case, 21 is the solution because substituting 21 for $w$ in the equation results in a true statement. 

$\begin{aligned}2(15) + 2w &=72\\ 2(15)+2(21) &= 72\\ 30 + 42 &=72\\ 72&=72 \end{aligned}$

An equation that contains two unknown quantities or two quantities that vary is called an equation in two variables. A solution to such an equation is a pair of numbers that makes the equation true. 

Suppose Tyler spends $45 on T-shirts and socks. A T-shirt costs $10 and a pair of socks costs $2.50. If $t$ represents the number of T-shirts and $p$ represents the number of pairs of socks that Tyler buys, we can can represent this situation with the equation:

$10t + 2.50p = 45$

This is an equation in two variables. More than one pair of values for $t$ and $p$ make the equation true.

In this situation, one constraint is that the combined cost of shirts and socks must equal $45. Solutions to the equation are pairs of $t$ and $p$ values that satisfy this constraint.

Combinations such as $t=1$ and $p = 10$ or $t=2$ and $p=7$ are not solutions because they don’t meet the constraint. When these pairs of values are substituted into the equation, they result in statements that are false.