Chapter 0.1, Problem 54E

(a)

To determine

To find: Solve the equations simultaneously by an algebraic method, either substitution or elimination. Write the conclusion.

Answer to Problem 54E

The system has no solution.it is an inconsistent system.

Explanation of Solution

Given information:

The given system of equation is $2x-5y=3,6x-15y=9$ .This system of equation is dependent and inconsistent.

Concept used: Elimination method to find the solution of system is used.

Calculation:

The given equations are:

  $\begin{array}{l}2x-5y=3\mathrm{...}(1)\\ 6x-15y=\mathrm{9....}\left(2\right)\end{array}$

Multiply equation (1) by $3$ it gives:

  $6x-15y=\mathrm{9.....}(3)$

Now subtract equation $\left(2\right)$ from $(3)$

  $\begin{array}{l}6x-6x+15y-15y=9-9\\ 0+0=0\\ 0=0\end{array}$

Elimination gives an equation that is always true The two equation in the system represents the same line. This is a dependent system that has infinite solutions.

Now check the condition of inconsistency that is if there is a system of equation

  $\begin{array}{l}{a}_{1}x+b_{1}y=c\\ a_{2}x+b_{2}y=c\end{array}$

Then the system is consistent and represent same line if:

  $\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$

Check this condition with the given system of equation it gives:

  $\begin{array}{l}\frac{2}{6}=\frac{-5}{-15}=\frac{3}{9}\\ \frac{1}{3}=\frac{1}{3}=\frac{1}{3}\end{array}$

The system of equation satisfies the above condition of consistency hence the system is dependent and has infinite many solutions.

(b)

To determine

To find: The solution of the equation of system by graph.

Answer to Problem 54E

The system is consistent

Explanation of Solution

Given information:

The given system of equation is $2x-5y=3,6x-15y=9$ .This system of equation is dependent and inconsistent.

Calculation:

To draw the graph of the equation $2x-5y=3,6x-15y=9$ . Substitute different values of $x$ to find the corresponding values of $y$ .

The graph of the above equation is given below.

  

As it is clear from the graph that two lines are coincide hence there is infinite number of solution of this system of equation has no solution.