Chapters

Equalization Method
Proposed Exercises for the Equalization Method

The equalization method is an effective technique for solving systems of linear equations, especially those involving two variables. This method is based on isolating one of the variables in both equations and then equalizing the expressions obtained. This allows us to find a specific value for one of the variables, which can then be substituted to find the value of the other.

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Equalization Method

The equalization method is based on the principle of transitivity.
If $\text{[math]}$ and then $\text{[math]}$ , then, by transitivity we know that $\text{[math]}$ .

Example:

If $\text{[math]}$ and we know that $\text{[math]}$ , then we can state that $\text{[math]}$ .
The same occurs in a system of equations using this method, as shown below.
Step 1: We select a variable that exists in each of the equations of the system.
Step 2: We isolate the variable in each of the equations.

Example:

$\text{[math]}$

We can isolate either of the 2 variables, in this case we have chosen $\text{[math]}$ . Remember to do it in each of the equations.

$\text{[math]}$
$\text{[math]}$

We can observe that both equations are equated with $\text{[math]}$ , so by transitivity we say that:

If $\text{[math]}$ and $\text{[math]}$ , then $\text{[math]}$ .

We can observe that now we only have one equation with one variable left, which we can simplify and isolate, obtaining:

$\text{[math]}$
$\text{[math]}$
$\text{[math]}$
$\text{[math]}$
$\text{[math]}$ $\text{[math]}$

Now we substitute the value of y in either of the 2 equations to obtain the value of $\text{[math]}$

$\text{[math]}$
$\text{[math]}$
$\text{[math]}$
$\text{[math]}$

Proposed Exercises for the Equalization Method

Score:

$\text{[math]}$

Solution

To solve by equalization we must isolate some variable from both equations. In this case we will isolate $\text{[math]}$ . In the first equation we obtain:

$\text{[math]}$

While for the second equation we obtain:

$\text{[math]}$

Equating the equations, we have

$\text{[math]}$

so that

$\text{[math]}$

so $\text{[math]}$ .

Then, substituting $\text{[math]}$ in the second equation, we have

$\text{[math]}$

so $\text{[math]}$ .

Thus, the solution is $\text{[math]}$ and $\text{[math]}$ .

$\text{[math]}$

Solution

As in the previous case, to solve by equalization we must isolate some variable from both equations. In this case we will isolate $\text{[math]}$ . In the first equation we obtain:

$\text{[math]}$

While for the second equation we obtain:

$\text{[math]}$

Equating the equations, we have

$\text{[math]}$

so $\text{[math]}$ .

Then, substituting $\text{[math]}$ in the first equation, we have

$\text{[math]}$

so $\text{[math]}$ .

Thus, the solution is $\text{[math]}$ and $\text{[math]}$ .

$\text{[math]}$

Solution

We isolate the unknown x from the first and second equation

$\text{[math]}$
$\text{[math]}$

$\text{[math]}$

We equate both expressions

$\text{[math]}$

We solve the equation

$\text{[math]}$

We substitute the value of $\text{[math]}$ , in one of the two expressions in which we have isolated $\text{[math]}$ .

$\text{[math]}$

Solution

We isolate the unknown $\text{[math]}$ from the first and second equation

$\text{[math]}$

We equate both expressions

$\text{[math]}$

We solve the equation

$\text{[math]}$

We substitute the value of y, in one of the two expressions in which we have isolated $\text{[math]}$

$\text{[math]}$

Solution

We isolate the unknown $\text{[math]}$ from the first and second equation.

$\text{[math]}$

We equate both expressions and solve the equation

$\text{[math]}$

We substitute the value of y, in one of the two expressions in which we have isolated $\text{[math]}$ .

$\text{[math]}$

Solution

We multiply the second equation by 2, to simplify it:

$\text{[math]}$

We arrange the terms

$\text{[math]}$

We isolate the unknown x from the first and second equation

$\text{[math]}$

We equate both expressions and solve the equation

$\text{[math]}$

We substitute the value of $\text{[math]}$ , in one of the two expressions in which we have isolated $\text{[math]}$ .

$\text{[math]}$

Solution

We clear denominators

$\text{[math]}$

We arrange the second equation

$\text{[math]}$

We isolate the unknown x from the first and second equation

$\text{[math]}$

We equate both expressions

$\text{[math]}$

We solve the equation

$\text{[math]}$

We substitute the value of $\text{[math]}$ , in one of the two expressions in which $\text{[math]}$ .

$\text{[math]}$

Solution

We isolate the unknown x from the first and second equation

$\text{[math]}$

We equate both expressions

$\text{[math]}$

We solve the equation

$\text{[math]}$

We substitute the value of $\text{[math]}$ , in one of the two expressions in which we have isolated $\text{[math]}$ .

$\text{[math]}$

$\text{[math]}$

Solution

Before applying the equalization method, we must write the system in a way that allows us to isolate one of the variables. To do this, we multiply both equations by 2:

$\text{[math]}$

We isolate the variable y in both equations:

$\text{[math]}$

Equating the equations, we have

$\text{[math]}$

thus

$\text{[math]}$

so $\text{[math]}$ . Then, substituting $\text{[math]}$ in the first equation, we have

$\text{[math]}$

so $\text{[math]}$ . Thus, the solution is $\text{[math]}$ and $\text{[math]}$ .

$\text{[math]}$

Solution

First we isolate $\text{[math]}$ from both equations

$\text{[math]}$

Equating the equations, we have

$\text{[math]}$

so that

$\text{[math]}$

so $\text{[math]}$ . Then, substituting $\text{[math]}$ in the second equation, we have

$\text{[math]}$

so $\text{[math]}$ . Thus, the solution is $\text{[math]}$ and $\text{[math]}$ .

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Agostina Babbo is an English and Italian to Spanish translator and writer, specializing in product localization, legal content for tech, and team sports—particularly handball and e-sports. With a degree in Public Translation from the University of Buenos Aires and a Master's in Translation and New Technologies from ISTRAD/Universidad de Madrid, she brings both linguistic expertise and technical insight to her work.

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