How to Properly Use Substitution

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SUMMARY

The discussion centers on the proper use of substitution in solving systems of equations. It highlights that substituting one variable without consistently applying it across all instances can lead to incorrect conclusions, as demonstrated with the equations x = x² and x² = 4. The incorrect substitution resulted in the erroneous equation x = 4, which does not reflect the original system's solutions. The consensus is that valid substitution requires replacing all occurrences of a variable to maintain the integrity of the equations.

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  • Understanding of algebraic equations and their solutions
  • Familiarity with substitution methods in algebra
  • Knowledge of variable representation and consistency in equations
  • Ability to solve quadratic equations
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  • Study the principles of substitution in algebraic equations
  • Learn about solving systems of equations using substitution
  • Explore common pitfalls in variable substitution
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Students learning algebra, educators teaching mathematics, and anyone interested in mastering the techniques of solving equations through substitution.

FAS1998
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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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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.
 
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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