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Explanation of The Substitution Method for Systems of Equations
Example of the Substitution Method
Proposed Exercises for The Substitution Method

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Explanation of The Substitution Method for Systems of Equations

The substitution method, as its name suggests, consists of substituting the value of a variable obtained from one of the equations and substituting it into the other equation. Systems of equations have a very important characteristic or rule: When a system of equations has more unknowns (variables) than the number of equations, then the system has infinite solutions, meaning each variable can take different values that always satisfy the equation, and the number of values each variable can take is infinite.

Given the equation

$\text{[math]}$

We observe that this is one equation with two variables. We can quickly realize some of the values that are solutions:

$\text{[math]}$

Note that there exists an infinite number of values we can assign to $\text{[math]}$ and $\text{[math]}$ for them to be solutions.

On the other hand, when the system has more equations than unknowns, then the system has a unique solution.

Example of the Substitution Method

$\text{[math]}$

Equation I: $\text{[math]}$

Equation II: $\text{[math]}$

We solve for either of the 2 variables in one of the 2 equations (we should always look for the one that requires less algebraic work for our convenience). In this case, we will solve for $\text{[math]}$ in Equation I.

$\text{[math]}$

This is called "Value of $\text{[math]}$ with respect to $\text{[math]}$ "

We substitute the solved value into the other equation. In this case, we substitute the value of $\text{[math]}$ into Equation II

$\text{[math]}$

As we can notice, now the equation only contains the variable $\text{[math]}$ . This equation can be simplified and solved to obtain the value of $\text{[math]}$ .

$\text{[math]}$

Once we have the value of one of the variables, in this case $\text{[math]}$ , we can substitute it into either of the 2 equations to find the value of the other variable, in this case $\text{[math]}$ .

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

And thus we obtain the value of our variables in a system of equations and notice that the solution is UNIQUE.

Proposed Exercises for The Substitution Method

Score:

$\text{[math]}$

Solution

We solve for $\text{[math]}$ in the second equation and simplify by dividing by 2 $\text{[math]}$

We substitute the value of variable $\text{[math]}$ in the other equation and solve the equation $\text{[math]}$

We substitute the value of $\text{[math]}$ in the second equation $\text{[math]}$

Therefore, the solution to the system of equations is $\text{[math]}$

$\text{[math]}$

Solution

We solve for $\text{[math]}$ from the second equation $\text{[math]}$

We substitute variable $\text{[math]}$ in the other equation and solve the equation $\text{[math]}$

We substitute the value of $\text{[math]}$ in the second equation $\text{[math]}$

Therefore, the solution to the system of equations is $\text{[math]}$

$\text{[math]}$

Solution

We eliminate denominators in the first equation by multiplying by 2 and rearrange the second $\text{[math]}$

We solve for $\text{[math]}$ in the second equation $\text{[math]}$

We substitute $\text{[math]}$ in the other equation $\text{[math]}$

We substitute the value of $\text{[math]}$ in the solved $\text{[math]}$ $\text{[math]}$

Therefore, the solution to the system of equations is $\text{[math]}$

$\text{[math]}$

Solution

We eliminate denominators $\text{[math]}$

We operate in the second equation $\text{[math]}$

We solve for $\text{[math]}$ in the first equation $\text{[math]}$

We substitute in the second equation and solve the equation $\text{[math]}$

We substitute the value of $\text{[math]}$ in the first equation $\text{[math]}$

Therefore, the solution to the system of equations is $\text{[math]}$

$\text{[math]}$

Solution

We solve for $\text{[math]}$ in the first equation $\text{[math]}$

We substitute the value of $\text{[math]}$ in the other equation and solve the equation $\text{[math]}$

We substitute the value of $\text{[math]}$ in the first equation $\text{[math]}$

Therefore, the solution to the system of equations is $\text{[math]}$

$\text{[math]}$

Solution

We solve for $\text{[math]}$ in the first equation $\text{[math]}$

We substitute the value of $\text{[math]}$ in the other equation and solve the equation $\text{[math]}$

We substitute the value of $\text{[math]}$ in the first equation $\text{[math]}$

Therefore, the solution to the system of equations is $\text{[math]}$

$\text{[math]}$

Solution

We solve for $\text{[math]}$ in the first equation $\text{[math]}$

We substitute the value of $\text{[math]}$ in the other equation and solve the equation $\text{[math]}$

We substitute the value of $\text{[math]}$ in the first equation $\text{[math]}$

Therefore, the solution to the system of equations is $\text{[math]}$

$\text{[math]}$

Solution

We solve for $\text{[math]}$ in the first equation $\text{[math]}$

We substitute the value of $\text{[math]}$ in the other equation and solve the equation $\text{[math]}$

We substitute the value of $\text{[math]}$ in the first equation $\text{[math]}$

Therefore, the solution to the system of equations is $\text{[math]}$

$\text{[math]}$

Solution

We solve for $\text{[math]}$ in the first equation $\text{[math]}$

We substitute the value of $\text{[math]}$ in the other equation and solve the equation $\text{[math]}$

We substitute the value of $\text{[math]}$ in the first equation $\text{[math]}$

Therefore, the solution to the system of equations is $\text{[math]}$

$\text{[math]}$

Solution

We solve for $\text{[math]}$ in the first equation $\text{[math]}$

We substitute the value of $\text{[math]}$ in the other equation and solve the equation $\text{[math]}$

We substitute the value of $\text{[math]}$ in the first equation $\text{[math]}$

Therefore, the solution to the system of equations is $\text{[math]}$

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Agostina

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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