Suppose we bought 2 packs of markers and a $0.50 glue stick for $6.10. If $p$ is the dollar cost of 1 pack of markers, the equation $2p +0.50 = 6.10$ represents this purchase. The solution to this equation is 2.80.

Now suppose a friend bought 6 of the same packs of markers and 3 $0.50 glue sticks, and paid $18.30. The equation $6p + 1.50 = 18.30$ represents this purchase. The solution to this equation is also 2.80.

We can say that $2p + 0.50= 6.10$ and $6p + 1.50 = 18.30$ are equivalent equations because they have exactly the same solution. Besides 2.80, no other values of $p$ make either equation true. Only the price of $2.80 per pack of markers satisfies the constraint in each purchase.

In this example, the second equation, $6p + 1.50 =18.30$, is a result of multiplying each side of the first equation by 3. Buying 3 times as many markers and glue sticks means paying 3 times as much money. The unit price of the markers hasn't changed.

Here are some other equations that are equivalent to $2p + 0.50= 6.10$, along with the moves that led to these equations.

In each case:

• The move is acceptable because it doesn't change the equality of the two sides of the equation. If $2p + 0.50$ has the same value as 6.10, then multiplying $2p + 0.50$ by $\frac12$ and multiplying 6.10 by $\frac12$ keep the two sides equal.
• Only $p=2.80$ makes the equation true. Any value of $p$ that makes an equation false also makes the other equivalent equations false. (Try it!)

These moves—applying the distributive property, adding the same amount to both sides, dividing each side by the same number, and so on—might be familiar because we have performed them when solving equations. Solving an equation essentially involves writing a series of equivalent equations that eventually isolates the variable on one side.

Not all moves that we make on an equation would create equivalent equations, however!

For example, if we subtract 0.50 from the left side but add 0.50 to the right side, the result is $2p = 6.60$. The solution to this equation is 3.30, not 2.80. This means that $2p = 6.60$ is not equivalent to $2p + 0.50 =6.10$.