How to Properly Use Substitution

In summary, the proper use of substitution in a set of equations results in a new equation with solutions that are also solutions of both of the previous equations. However, if done incorrectly, such as substituting for one variable but not the other, the resulting equation may be incorrect. It is important to use different names for different variables and to substitute for all instances of a variable. In this particular case, substituting ##x=\sqrt{2}## would show that there are no solutions to the system of equations.
  • #1
FAS1998
50
1
When substitution is properly used for a set of equations, I believe you get a new equation with solutions that are also solutions of both of the previous equations.

The following equation has solutions x = 0 and x = 1.

##x=x^2##

This next equation has solutions x = -2 and x = 2.

##x^2=4##

but if we replace x^2 (in the second equation) with x from the first equation, we get the equation

##x=4##

which is incorrect. I believe correct use of substitution would should that there are no solutions to the system of equations.

Why is this substitution incorrect, and when can we validly substitute parts of one equation into another equation.
 
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  • #2
Normally when using substitution of one equation into another we solve for some varaible x and sub it in the second wherever x occurs.

It looks like you subbed for x^2 but not for x and you must do it for all the places where x occurs.

In this case subbing in ##x=\sqrt 2## too.
 
  • #3
FAS1998 said:
I believe correct use of substitution would should that there are no solutions to the system of equations.
No! The correct way is not to use the same name for different things! You will not write banana on your list if you mean to buy apples. I cannot see why you do it here, and why we even discuss on this level!

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1. What is substitution and why is it important in scientific research?

Substitution is the act of replacing one variable or element with another in a scientific equation or experiment. It is important in scientific research because it allows for the manipulation and control of variables, which is crucial for accurately testing hypotheses and drawing conclusions.

2. How do I know when to use substitution in my experiments?

Substitution is typically used when there is a need to change or manipulate a variable in an experiment. This can be determined by identifying the independent and dependent variables in your research question or hypothesis. If you need to change the independent variable, substitution may be necessary.

3. Can substitution be used in all types of scientific experiments?

Yes, substitution can be used in a variety of scientific experiments, including chemistry, physics, biology, and more. It is a fundamental concept in the scientific method and is applicable in many different fields of study.

4. What are some common mistakes to avoid when using substitution?

One common mistake is substituting the wrong variable or value, which can lead to inaccurate results. It is important to double-check and carefully follow the instructions for substitution in your experiment. Another mistake is using substitution without fully understanding the concept, which can lead to confusion or errors in data analysis.

5. Are there any alternative methods to substitution in scientific research?

Yes, there are other methods that can be used to manipulate variables in scientific experiments, such as randomization or control groups. However, substitution is a widely used and effective method that allows for precise control and manipulation of variables, making it a valuable tool in scientific research.

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