We can use the relationship between addition and subtraction to reason about subtracting signed numbers. For example, the equation $7 - 5 = {?}$ is equivalent to $5 + {?} = 7$. Here is a diagram that represents the addition equation.

A number line with the numbers negative 10 through 10, indicated. An arrow starts at 0, points to the right, ends at 5, and is labeled "plus 5." A second arrow starts at 5, points to the right, ends at 7, and is labeled with a question mark. There is a solid dot indicated at 7.

To get to the sum of 7, the second arrow must be 2 units long, pointing to the right. This tells us that positive 2 is the number that completes each equation: $5 + 2 = 7$ and $7 - 5 = 2$.

Notice that the addition expression $7 + (\text-5)$ also equals 2.

A number line with the numbers negative 10 through 10, indicated. An arrow starts at 0, points to the right, ends at 7, and is labeled "plus 7". A second arrow starts at 7, points to the left, ends at 2, and is labeled "minus 5". There is a solid dot and a question mark labeled at 2.

So we can see that $7 - 5 = 7 + (\text-5)$.

Here's another example. The equation $3 - 5 = {?}$ is equivalent to $5 + {?} = 3$.

A number line with the numbers negative 10 through 10, indicated. An arrow starts at 0, points to the right, ends at 5, and is labeled "plus 5." A second arrow starts at 5, points to the left, ends at 3, and is labeled with a question mark. There is a solid dot indicated at 3.

To get the to the sum of 3, the second arrow must be 2 units long, pointing to the left. This tells us that -2 is the number that completes each equation: $5 + \text-2 = 3$ and $3 - 5 = \text-2$.

Notice that the addition expression $3 + (\text-5)$ also equals -2.

A number line with the numbers negative 10 through 10 indicated. An arrow starts at 0, points to the right, ends at 3, and is labeled "plus 3". A second arrow starts at 3, points to the left, ends at negative two, and is labeled "minus 5". There is a solid dot and a question mark labeled at 2.

So we can see that $3 - 5 = 3 + (\text-5)$.

This pattern always works. In general:

$a - b = a + (\text-b)$