Chapter 6.1, Problem 3P

a.

To determine

To find: the solution of given pair of linear equations by graphing.

Answer to Problem 3P

No solution

Explanation of Solution

Given information: a pair of linear equation is given as follows-

  $\begin{array}{l}y=-x-3\\ y=-x+5\end{array}$

Graph:

  

Interpretation: since the graph of both the linear equation is parallel to each other and they will never cross each other.

Hence there is no solution for given pair of linear equations.

b.

To determine

To find: the solution of given pair of linear equations by graphing.

Answer to Problem 3P

Infinitely many solutions

Explanation of Solution

Given information: a pair of linear equation is given as follows-

  $\begin{array}{l}y=3x-3\\ 3y=9x-9\end{array}$

Graph:

  

Interpretation: since the graph of both the linear equations is same and coincide each other. Hence there is infinitely many solutions for given pair of linear equations.

c.

To determine

To state: how can we determine without graphing whether given system of equations has one solution, infinitely many solutions, or no solution?

Explanation of Solution

let a system of equations be as follows-

  $\begin{array}{l}{a}_{1}x+b_{1}y=c_1\\ a_{2}x+b_{2}y=c_2\end{array}$

Then the given system of equations will have only one solution if $\frac{a_1}{a_2}\ne \frac{b_1}{b_2}$ .

The given system of equations will have infinitely many solutions if $\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$ .

The given system of equations will have no solution if $\frac{a_1}{a_2}=\frac{b_1}{b_2}\ne \frac{c_1}{c_2}$ .