Can You Solve These Linear Simultaneous Equations?

  • Thread starter isaac200
  • Start date
  • Tags
    Linear
In summary, the conversation discusses two possible plans for solving a set of equations involving x, y, and z. Plan A involves multiplying out equations and using the values of yz + zx + xy and xyz to write a cubic equation. Plan B involves guessing that the solution will be integers and finding three integers whose squares sum to 35. The conversation also explores the potential importance of the order of the integers and the relevance of posting in the "Mechanical Engineering" category. Finally, the solution is explained using the substitution method, resulting in the values of x, y, and z being 5, -1, and -3.
  • #1
isaac200
7
0
Can any solve this x+y+z=1 x^2+y^2+z^2=35 x^3+y^3+z^3=97
 
Engineering news on Phys.org
  • #2
Plan A: multiply out (x+y+z)^2 and (x+y+z)^3.
From that and the given eqations, you can get the values of yz + zx + xy and xyz.
Then, you can write down a cubic equation whose roots are x y and z.

Plan B: take a guess that the solution will probably be integers, and find 3 integers whose squares sum to 35. If you don't get lucky, try plan A :smile:
 
  • #3
5, -3 & -1 work... now about there order?
 
  • #4
You are correctbut why don't you show your working?
 
  • #5
x+y+z=1 x^2+y^2+z^2=35 x^3+y^3+z^3=97

Honestly, I just "guess and checked".

5+(-3)+(-1)=1
5^2+(-3)^2+(-1)^2=35
5^3+(-3)^3+(-1)^3=97

There isn't a way to find the order as far as I can see.
 
  • #6
huntoon said:
5, -3 & -1 work... now about there order?

huntoon said:
There isn't a way to find the order as far as I can see.

Why do you think order would matter here? And, why is this posted in "Mechanical Engineering"?
 
  • #7
Am sorry that this is post in the wrong position but this is the solution x^2+y^2+z^2=(x+y+z)^2-2(xy+yz+zx)=35 substitute x+y+z=1 we have (1)^2-2(xy+yz+zx)=35 therefore (xy+yz+zx)=-17
also x^3+y^3+z^3=(x+y+z)^3-3(x+y+z)(xy+yz+zx)+3(xyz)=97
therefore substitute we have
(1)^3-3(1)(-17)+3(xyz)=97 which give (xyz)=15 now factor of 15 whose sum is 1 give 5,-1,-3
 

1. What is a linear simultaneous equation?

A linear simultaneous equation is a mathematical equation that contains two or more variables and can be expressed as a system of linear equations. This means that the highest power of any variable in the equation is 1. These equations can be solved simultaneously to find the values of the variables that satisfy all of the equations.

2. How is a linear simultaneous equation solved?

A linear simultaneous equation is typically solved using techniques such as substitution or elimination. These methods involve manipulating the equations to isolate one variable and then substituting its value into the other equations to find the remaining variables. Another method is using matrices and Gaussian elimination.

3. What is the importance of linear simultaneous equations in science?

Linear simultaneous equations are important in science because they can be used to model and solve real-world problems that involve multiple variables. They are commonly used in physics, chemistry, economics, and other fields to analyze and predict various phenomena.

4. Can a linear simultaneous equation have no solution?

Yes, it is possible for a linear simultaneous equation to have no solution. This occurs when the equations are inconsistent, meaning that there is no set of values that satisfy all of the equations. Graphically, this would represent two parallel lines that do not intersect.

5. How do I know if a linear simultaneous equation has infinitely many solutions?

If the equations in a linear simultaneous equation are consistent, meaning that they have at least one solution, and the number of variables is equal to the number of equations, then the system will have a unique solution. If the number of variables is greater than the number of equations, then the system will have infinitely many solutions. This can be visualized as two lines that are identical or overlapping.

Similar threads

  • Mechanical Engineering
Replies
8
Views
795
Replies
1
Views
811
  • Calculus and Beyond Homework Help
Replies
7
Views
642
  • Calculus and Beyond Homework Help
Replies
1
Views
416
  • Mechanical Engineering
Replies
3
Views
365
  • Precalculus Mathematics Homework Help
Replies
11
Views
798
  • Calculus and Beyond Homework Help
Replies
8
Views
348
  • Mechanical Engineering
Replies
7
Views
2K
Replies
9
Views
1K
  • General Math
Replies
2
Views
687
Back
Top