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# Cancel|Definition & Meaning

## Definition

The cancel **operation** or the process of **cancellation** is a mathematical way in which **smaller expressions** or factors are removed from **larger expressions** or numerals. This cancellation process may be due to **addition/subtraction** or due to **any other operation** that may be the case.

When we talk about the **cancel operation** or the process of **cancellation** from the perspective of **mathematics,** we can’t really find much material for its proper formal **introduction.** This article will shed light on this important **process** in line with textbook **methods.** We will see how this process is **important** and how it may appear in different **roles** and different **types** of problems.

**Figure 1:** Graph of the **Line** **2x + 3y + 5 = 0 **

The **simplest process** of cancellation may be based on **addition** or **subtraction** operations. However the operation is not limited to these **arithmetic phenomena** only. Rather the cancellation process may be applied to any other **complex** or **implicit** function as long as the **inverse operator** is defined.

In the following paragraphs, we explain the **concept** in more detail.

## Examples of Cancellation Process

There is a **special type** of cancellation process called the **anomalous cancellation.** Anomalous cancellation is such a process in which given a **fraction** of the form **p/q,** we may write it in the form of **ac/bd** such that **a = b** which further **reduces** the expression to **c/d.** Now **c/d** is a **simpler expression** that may look **different** yet the **mathematical meaning** remains the **same.**

For **example,** consider the case of following **fraction:**

**30/70**

This **expression** may be written as:

**30/70 = (3/10) / (7/10) = 3/7**

Now, the fraction **3/7** is a much **simpler expression,** but it has the **same meaning** in terms of mathematics.

## Visual Intuition of Cancel Operation

To understand the **cancel operation** we need to consider an **example.** Let us say that you have an equation of a **line** defined as follows:

**2x + 3y + 5 = 0**

Keep in mind that it is derived from the **standard line equation** **( ax + by + c = 0 ).** The figure 1 plots this line in a 2D XY plane. Now let us **multiply** this equation with a **scalar,** say **10,** for example, and see how the line is affected. The above equation will change as follows:

**10 (2x + 3y + 5) = 10(0)**

**20x + 30y + 50 = 0**

Following **figure** plots this new line in 2D XY plane:

**Figure 2:** Graph of the **Line** **20x + 30y + 50 = 0**

If you compare **figures 1 and 2**, you can clearly see that there is **no difference** in the actual line and its graph. **Both** **equations** represent the **same line** in the physical world. This means that the factor we multiplied with the original equation had **no effect** on the **mathematical meaning** of the equation or **expression.**

If we **converse** the statement, we can **conclude** that starting from the **polynomial** **20x + 30y + 50 = 0,** we can find a factor that is **common** on the left-hand side, and it can be **canceled out** without any change in the **mathematical meaning** of the expression.

Consider another example of a **parabola.** Suppose you are given the **following equation** of a parabola:

**y + 10x + 7 = x ^{2} + 8x + 4**

The following **figure** plots this equation in the 2D plane:

**Figure 3:** Graph of the **Parabola** y + 10x + 7 = x^{2} + 8x + 4

Now at **first glance,** the equation looks nothing like a parabola and seems **quite complex.** If you look closely, it can be seen that the two of three expressions on the left hand side can be **canceled out** from the right hand side if we just **split** the right hand side like so:

**y + (2x + 8x) + (3 + 4) = x ^{2} + 8x + 4**

**y + 2x + 8x + 3 + 4 = x ^{2} + 8x + 4**

Now it is very clear that the terms **8x and 4** are on both sides of the equation and can be **canceled out,** which reduces the equation to the following:

**y + 2x + 3 = x ^{2}**

**Rearranging** the equation yields:

**y = x ^{2} – 2x – 3**

Which is a quadratic equation that looks much **more simpler** and similar to a **parabola equation.** The following figure plots the above **expression** along with the expression plotted in **figure 3.**

**Figure 4:** Graph of the **Parabola** y = x^{2} – 2x – 3

It is clear the the **simpler expression** of figure 4 and the more **complex form** of figure 3 are **mathematically the same.** These examples clarify the meaning and **significance** of the **cancellation process.** There are also quite a few simpler examples that we will solve in the following section of the article.

## Numerical Problems of Cancelation

**Simplify** the **following expressions** using the **cancellation** or anomalous cancellation:

**(a)** y + 15 x = 10 x + 5

**(b)** 144/108

**(c)** (55 * y) / 10 + 2 x = 55 / 10

### Solution

**Part (a) – **Given that:

**y + 15x = 10x + 5**

**Splitting** 15x into two terms as the sum of 10x and 5x:

y + 10x + 5x = 10x + 5

**Canceling out** the 10x term on both sides:

**y + 5x = 5**

which is the **simplest expression.**

**Part (b) – **We are given that:

**144/108**

**Factorizing** both the denominator and the numerator, we end up with the following:

(12)(12) / (12)(9)

**Canceling out** the factor 12 as it exists in both numerator and denominator:

12 / 9

Further **factorizing** the fraction, we get:

(4)(3) / (3)(3)

**Canceling out** the factor of 3 in the numerator and denominator:

**4 / 3**

which is the **simplest expression.**

**Part (c) – **We have the expression:

**(55 * y) / 10 + 2 x = 55 / 10**

**Factorizing** both the denominator and the numerator on both sides:

(11)(5)(y) / (2)(5) + 2 x = (11)(5) / (2)(5)

**Canceling out** the factor of 5 as it exists in both the numerator and denominator:

**(11 * y) / 2 + 2 x = 11 / 2**

which is the **simplest expression.**

*All mathematical drawings and images were created with GeoGebra.*