Question about this technique for solving simultaneous equations

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chwala
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Homework Statement
See attached
Relevant Equations
understanding of equations
I was going through this...

1693204391194.png


The steps are quite clear; although i do not know whether it is a general approach to let ##y=mx## in such kind of problems when the degree are the same...second degree, third degree and so on.

My approach to this problem was straightforward;

##y=\dfrac{8-2x^2}{3x}##

thus on substitution to first equation, we shall have,
...
##9x^4+96x^2-24x^4+64-32x^2+4x^4-117x^2=0##

##-11x^4-53x^2+64=0##

Let

##m=x^2##

then it follows that,

##11m^2+53m-64=0##

##m=1, ⇒ x=±1##

The values of ##y## would be found by substituting ##x=±1## into ##y=\dfrac{8-2x^2}{3x}##

cheers.

My interest is on the highlighted part.
 
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  • #2
As is clear from the working, ##m## is a variable, not a constant of proportionality. There was a thread yesterday where a similar approach caused this confusion. In general, as long as ##x \ne 0##, you can always set ##m = \frac y x##. Personally, I would use ##z = \frac y x##, and then it's clearer what's happening.
 
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chwala said:
My interest is on the highlighted part.
I don't see any highlighted part.
 
  • #4
Mark44 said:
I don't see any highlighted part.
I assume it is the passage in lilac: "although i do not know whether it is a general approach to let in such kind of problems when the degree are the same...second degree, third degree and so on."
 
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haruspex said:
I assume it is the passage in lilac: "although i do not know whether it is a general approach to let in such kind of problems when the degree are the same...second degree, third degree and so on."
You're right. I thought he meant that something was highlighted in the image from the book. Also, that lilac doesn't really stand out very distinctly.
 

Related to Question about this technique for solving simultaneous equations

What is the substitution method for solving simultaneous equations?

The substitution method involves solving one of the equations for one variable in terms of the other and then substituting this expression into the other equation. This reduces the system to a single equation with one variable, which can be solved directly. Once the value of one variable is found, it can be substituted back into one of the original equations to find the value of the other variable.

How does the elimination method work for solving simultaneous equations?

The elimination method involves adding or subtracting the equations to eliminate one of the variables. This is done by aligning the coefficients of one of the variables and then adding or subtracting the equations to cancel out that variable. This reduces the system to a single equation with one variable, which can be solved. The value of this variable is then substituted back into one of the original equations to find the value of the other variable.

When should I use the graphing method for solving simultaneous equations?

The graphing method is useful for visualizing the solutions of simultaneous equations. It involves plotting both equations on a coordinate plane and identifying the point(s) where the graphs intersect. This intersection point represents the solution to the system. This method is particularly useful for understanding the relationship between the equations, but it may not be precise if the intersection points are not clear.

What are the advantages and disadvantages of the matrix method for solving simultaneous equations?

The matrix method, or using linear algebra techniques like Gaussian elimination or Cramer's rule, is particularly powerful for solving large systems of equations. The advantages include systematic procedures and the ability to handle many equations simultaneously. However, the disadvantages are that it can be computationally intensive and may require a solid understanding of matrix operations and determinants.

Can simultaneous equations have no solution or infinitely many solutions?

Yes, simultaneous equations can have no solution or infinitely many solutions. If the equations represent parallel lines, they will never intersect, indicating no solution. If the equations represent the same line, they will intersect at infinitely many points, indicating infinitely many solutions. This is determined by the consistency and dependency of the equations in the system.

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