math help????????

Question

Write a system of three linear equations in three variables that has (2,1,5) as a solution. (There are many possibilities.) Explain the process you used to write an equation.

Answer 1

Hi Samantha,

  The solution is:

     x = 2

     y = 1

     z = 5

That’s three linear equations right there but they don’t yet have three variables each.  So, (including 0 as coefficients)

 2*x + 0*y + 0*z = 4

0*x + 1*y +0*z = 1

0*x + 0*y + 5*z = 5

are three lines (in 3D) that intersect at point (2,1,5).  As you mentioned, there are more possibilities because once we have three intersecting lines, we could simply move the x-y-z axis anywhere else that we wanted (sometimes transformation of data makes calculations much simpler).

This process uses the given point (2,1,5) to get the coefficients of x, y, and z and plugs in 0 for other coefficients.  (Note:  remember the term “matrix” because you will learn about that someday.)

Keep it simple!    Best wishes.

--d.w.

Answer 2

The simplest system is:

 

x = 2

y = 1

z = 5

 

For anything more complex the derivation is rather convoluted.

Answer 3

Based on the form of the ordered pairs, we will have 3 equations, since we have 3 variables.  We will use the order pairs to come with the equations.

 

 

The first one can be the sum of the ordered pairs.

 

x + y + z = 8          eq1

 

 

For the second one, we can add a coefficient greater than 1 to the variables. 

 

2x + y + z = 10      eq2

 

 

For the third one, we can do the same thing we did for the last one. 

 

4x + 5y - 2z = 3           eq3