How to Determine Values for a and b in a 3x4 System for Infinite Solutions?

In summary, using a sledgehammer to force the solution to fall out is important, but honing your skills at seeing a quick and easy way to solve the problem is just as important.
  • #1
Ewan_C
8
0
[Solved] 3x4 system of equations

Homework Statement



Consider the following system of three equations in x, y and z.

2x + 4y + 5z = 17
4x + ay + 3z = b
8x + 7y + 13z = 40

Give values for a and b in the second equation that make this system consistent, but with an infinite set of solutions.

The Attempt at a Solution



I found the answers a= -1, b=6 easily enough. I was told by my teacher that if a system of three equations has infinite solutions, one of the equations can be found from the other two. I multiplied equation 1 by 2 and subtracted the result from equation 3. This gave:

4x - y + 3z = 6

and so finding the values of a and b was pretty simple from there. Plugging the numbers into a calculator gave an infinite number of solutions.

My question is, how can I better explain how to get a and b from the provided data? My method just seems like an educated guess rather than solid evidence - I don't think it'd look very good to an examiner. Cheers.
 
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  • #2
Your method is perfectly good. What you have shown is that a combination of the first and third equation give the middle equation if a= -1 and b= 6. It was clever of you to notice that "twice the first equation subtracted from the first" gives an equation that can be made exactly the same as the second if a and b have those values. I see no "guessing" there.

What I might have done (perhaps because I am not as smart as you!) would be to try to solve the system of equations and see what might prevent me from getting a single solution. If I subtract twice the first equation from the second, the "x"s cancel and I am left with (a- 8)y- 7z= b- 34. If I subtract 4 times the first equation from the third, again the "x"s cancel and I am left with -9y- 7z= 28. Now I can eliminate the z by subtracting that last equation from the previous one: [(a-8)+ 9]y= (a+1)y= b- 6. If a+1 is not 0, I can divide by it and get a single answer. In order that there not be a single answer, we must have a+ 1= 0 or a= -1. Of course, that would make the equation 0= b- 6. In order that that be true (for any y) we must have b= 6.

Frankly, I prefer your method.
 
  • #3
Ewan, welcome to PhysicsForums. Halls: Nice humble response. I'm of Halls' mind here. I too tend to use a sledge hammer to force the solution to fall out where a little poking around would make the solution fall out with little effort. One of the PF 2007 guru award winners, Dick, is very good at poking around and making the solution fall out with little effort. (He also knows how to wield sledge hammers when needed.) Honing your skills with the heavy-handed techniques is important, but so is honing your skills at seeing a quick and easy way to solve the problem. There's nothing wrong with a bit of creativity.
 
  • #4
Thanks for the responses. I guess I'll stick with my method in the OP then.
 

Related to How to Determine Values for a and b in a 3x4 System for Infinite Solutions?

1. What is a 3x4 system of equations?

A 3x4 system of equations is a set of three equations with four variables. Each equation contains three of the variables, and the goal is to find values for all four variables that satisfy all three equations simultaneously.

2. How many solutions can a 3x4 system of equations have?

A 3x4 system of equations can have zero, one, or infinitely many solutions. The number of solutions depends on the values of the coefficients and constants in the equations.

3. How do you solve a 3x4 system of equations?

To solve a 3x4 system of equations, you can use a variety of methods such as substitution, elimination, or Gaussian elimination. The specific method used will depend on the complexity of the equations and personal preference.

4. Can a 3x4 system of equations be inconsistent?

Yes, a 3x4 system of equations can be inconsistent, meaning there is no set of values for the variables that satisfy all three equations. This can occur if the equations are contradictory or if there are more variables than equations.

5. What real-world problems can be represented by a 3x4 system of equations?

A 3x4 system of equations can represent a variety of real-world problems, such as determining the optimal combination of ingredients for a recipe, finding the dimensions of a rectangular prism with a given volume and surface area, or calculating the intersection point of three lines in a three-dimensional space.

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