Chapter 6.2, Problem 4P

To determine

To calculate: number of solutions of the given system of equations.

Answer to Problem 4P

Infinite solutions

Explanation of Solution

Given information: Equations are $6y+5x=8,2.5x+3y=4$

Formula used: For the system of equations $a_{1}x+b_{1}y+c_1=0,a_{2}x+b_{2}y+c_2=0$ , if $\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$ , the system of equations has infinite solutions.

Calculation:

Consider the equation $6y+5x=8$

This equation can be written as $5x+6y-8=0$

Compare this equation with $a_{1}x+b_{1}y+c_1=0$

  $a_1=5,b_1=6,c_1=-8$

Consider the equation $2.5x+3y=4$

This equation can be written as $2.5x+3y-4=0$

Compare this equation with $a_{2}x+b_{2}y+c_2=0$

  $a_2=2.5,b_2=3,c_2=-4$

Find ratios $\frac{a_1}{a_2},\frac{b_1}{b_2},\frac{c_1}{c_2}$

  $\begin{array}{l}\frac{a_1}{a_2}=\frac{5}{2.5}=\frac{50}{25}=2\\ \frac{b_1}{b_2}=\frac{6}{3}=2\\ \frac{c_1}{c_2}=\frac{-8}{-4}=2\end{array}$

As $\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$ , the system of equations has infinite solutions.