Chapter 3.1, Problem 66PPS

To determine

To solve: The given system of equations

Answer to Problem 66PPS

The system has infinitely many solutions .

Explanation of Solution

Given Information:

The system of equations

  $\begin{array}{l}\frac{3}{2}y+z=3\\ -y-\frac{2}{3}z=-2\end{array}$

Formula Used:

Here, two equations with two unknown variables are given. The approach used is variable elimination and simplification.

(1) Bothequations are multiplied with a factor such that one of the variables has identical coefficients.

(2) Add the modified equations to eliminate the particular variable and solve for the other variable.

(3) Substitute the obtained variable in one of the equations to get the remaining variable.

Calculation:

Here, the equations are,

  $\frac{3}{2}y+z=3$........(1)

  $-y-\frac{2}{3}z=-2$........(2)

Multiplying each of the terms in equation (2) by $\frac{3}{2}$ , it becomes

  $-\frac{3}{2}y-\left(\frac{2}{3}\right)\left(\frac{3}{2}\right)z=-2\left(\frac{3}{2}\right)$

  $-\frac{3}{2}y-z=-3$........(3)

Now, adding corresponding terms of (3) and (1) gives,

  $\{\begin{array}{l}\frac{3}{2}y+z=3\\ -\frac{3}{2}y-z=-3\end{array}$

0y+0z=0

This indicates that there are infinitely many solutions for the given system of equations.

For a given value of y, the value of z can be expressed using (1) as,

  $z=3-\frac{3}{2}y$

Thus, the solution for the given system of equations can be expressed as,

  $(y,z=3-\frac{3}{2}y)$