# For the following system of equations in echelon form, tell how many solutions there are in non...

## Question:

For the following system of equations in echelon form, tell how many solutions there are in non negative integers.

x + 3y + z = 92

7y + 2z = 56

Select the correct choice below and, if necessary, fill in the answer box to complete your choice.

A. There are {eq}__ {/eq} non negative solutions.

B. There are infinitely many solutions.

C. There is no solution.

## Solution of a System of Equations :

For a system of equations if the coefficient determinant is nonzero then there exists a unique nonzero solution.

If number of equations is less than the number of variables then there exist either no solution or infinitely many solutions.

## Answer and Explanation:

Here the system of equations is:

{eq}x+3y+z=92 {/eq}

{eq}7y+2z=56 {/eq}

{eq}7y=56-2z {/eq}

{eq}y=\dfrac{56-2z}{7} {/eq}

{eq}x=92-3y-z=92-\dfrac{3}{7}(56-2z)-z=\dfrac{476+5z}{7} {/eq}

The values of the variables x,y depends on the value of the variable z.

Here the number of equations ie 2 is less than the number of variables ie 3.

As the value of z varies, therefore the given system has infinitely many solutions.

Hence B is the correct option.

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from High School Algebra II: Homework Help Resource

Chapter 8 / Lesson 8