Equations Equivalence Principles

For some types of equations, finding the solution is a straightforward process. However, for more complex equations, finding a solution might not be so obvious.

In these cases, the concept of an Equivalent Equation comes to our aid. That is, one equation is equivalent to another if they have the same solution. This concept is of paramount importance because if we can transform a complex equation into a simpler one, we have effectively found a way to solve the initial equation.

For this reason, the Equations Equivalence Principles exist, which allow us to move from one equation to another equivalent to it. The process of solving an equation is based precisely on these two principles.

In this lesson, we will study the principles of equivalence in detail.

Equivalent Equations

In the previous lesson, we saw that the solutions to an equation form a set referred to as the set of solutions and is often denoted by .

Two equations can have the same set of solutions, that is, they can be verified by the same solutions. Let's consider the following two equations:

Both of the above equations have the single solution . In fact, if we substitute this value into the two equations, we get:

In such a case, it's said that the two equations are equivalent.

Let's look at some examples.

Consider another example:

The concept of equivalent equations underpins the process of equation solving. In fact, to solve an equation, the aim is to transform the starting equation into other equivalent equations progressively simpler. The ultimate goal is to obtain an equation where finding the solution is immediate.

The mathematical notation used to indicate the transition from one equation to another equivalent one is an arrow . For instance:

Now that we have introduced the concept of equivalent equations, we can study the principles by which it is possible to transition from one equation to another equivalent one. These principles are called Equations Equivalence Principles.

The First Equation Equivalence Principle

We know from the laws of monotonicity that if we add the same amount to both sides of an equality, the equality remains valid.

From this law, we can derive the First Equation Equivalence Principle for equations.

To understand the first principle, let's start with a simple example. Consider the following equation:

This equation has a single solution: . In fact, substituting for the unknown, the equality is verified:

Applying the first principle, we can add the same quantity, a number or an expression, to both sides of the equation. Let's try adding the same number, , to each side:

Based on the first principle, this new equation is equivalent to the starting one. Therefore, this equation should have the same set of solutions: .

Let's verify that is indeed the solution. Substituting for the unknown, we get:

However, when adding an expression instead of a number, caution is needed. The first principle indeed states that the same expression can be added to both sides of an equation provided it's defined over the same domain.

In other words, when adding an expression, we need to ensure that the domain over which the expression is defined matches that of the equation. If, for instance, the equation is defined over the set of all real numbers, , we can't use an expression not defined over .

To illustrate, let's return to the earlier equation, which is clearly defined over all real numbers, and its solution is :

We can't add the expression to both sides of this equation. If we try, we get:

This equation, however, does not admit the solution . If the unknown assumes the value , the expression becomes undefined as it would transform into .

The mistake made in adding this expression comes from the fact that the equation is defined over all real numbers, , whereas the expression is defined in the set . Hence, the two domains don't match, and the first equivalence principle doesn't hold.

Conversely, we can add the expression since it's defined over all of . Doing so, we get:

Having adhered to the first principle, this equation will have the same set of solutions as the initial one. We can verify by substituting :

Thus, the resulting equation is equivalent to the starting equation.

Applications of the First Equation Equivalence Principle

Using the first equivalence principle, we can derive two helpful practical rules for solving equations.

Let's suppose we want to solve the following equation:

To solve this equation, we need to act in such a way as to reduce it to a simpler and immediate equation of the type:

where is a numerical value. In this way, the result is straightforward since the solution to the equation is precisely .

To achieve this, we must ensure that only the variable appears on the left side of the equation, while only a numerical value appears on the right side.

In the equation above, however, the variable appears on both sides. We must somehow eliminate the expression from the right side.

By applying the first principle, we can add the same expression to each side, like this:

We were able to perform this operation because the domain of the equation is all of , and the expression is defined over the entire set of real numbers .

Using this technique, we removed the variable from the right side, which now only appears on the left side. If observed closely, applying this technique is as if we have transported the expression from one side to the other but with the opposite sign.

Therefore, this rule is named the Transport Rule:

Returning to the previous equation, we can conclude the solution by again applying the transport rule to the term , moving it to the right side:

Thus, the solution to the equation is .

From the first equivalence principle, we can derive another useful rule for solving equations.

Consider the following example:

In this case, we must isolate the variable on the left side. To do this, we can add the term to both sides, eliminating the term :

As we can see, the terms on both sides disappeared, meaning we have canceled them.

From this, we can derive a second rule from the first principle, allowing us to cancel equal terms from both sides. This rule is aptly named the Cancellation Rule:

Let's apply the cancellation rule to another example:

In this equation, the term appears on both sides. Thus, we can apply the cancellation rule and remove this term from both sides:

The Second Equations Equivalence Principle

From the second law of monotony, we know that if we multiply both sides of an equality by the same quantity, the equality remains true.

From this law, we can derive the Second Equations Equivalence Principle:

To better understand the second principle, let's start with an example:

The solution to this equation is , indeed, by substituting for the variable we get:

We can apply the second principle to this equation. For instance, we can multiply each side by the numerical constant :

Let's check if the equation we derived is equivalent by substituting the solution of the first equation for the unknown: .

Hence, the two equations are equivalent.

Similarly, we could have divided the initial equation by a number, for example :

Let's verify if this equation is also equivalent to the original by substituting the solution for the unknown:

Thus, the resulting equation is equivalent.

When applying the second principle, one must be careful. Indeed, one cannot divide both sides of the equation by zero since division by zero is undefined.

Similarly, it doesn't make sense to multiply both sides by zero. Doing so would result in an equation that is not equivalent to the original.

Let's consider an example:

The solution to this equation is . If we try to multiply each side by zero, we get:

We have derived an equation where the solution can be any real number, so . In short, the resulting equation is not equivalent to the initial one.

Applications of the Second Equivalence Principle

As done for the first principle, we can derive simple rules for applying the second equivalence principle that can aid us in solving equations.

Let's start with an example.

The procedure applied in the above example can be generalized, thereby obtaining a new rule we can use to solve equations: the simplification rule.

Another significant rule we can derive from the second principle of equivalence is illustrated in the following example.

In the previous example, the pivotal moment was represented by the final step. Indeed, we multiplied both sides of the equation by . In doing so, we derived an equivalent equation where all signs of the terms were reversed. From this, we can extract a highly useful practical rule known as the sign changing rule.

A highly useful final application of the second equivalence principle concerns equations with non-integer coefficients. To understand better, let's examine the following example.

As the final step of the example demonstrates, converting an equation with fractional coefficients into one with integer coefficients simply involves multiplying both sides by the least common multiple of the denominators. This handy rule is the integer coefficients reduction rule:

In Summary

This lesson is pivotal as it introduces the concept of an equivalent equation, which underpins the equation-solving process. By utilising the equivalence principles, it's possible to transform one equation into a simpler one, and so on, until we arrive at an equation where finding a solution is straightforward.

From the two principles of equivalence, which directly derive from the monotonic laws of arithmetic operations, we've extracted five handy rules to simplify equations:

• The Transport Rule
• The Cancellation Rule
• The Simplification Rule
• The Sign Changing Rule
• The Integer Coefficients Reduction Rule

By employing the principles of equivalence alongside these five rules, we can proceed to solve equations.