Chapter 3.1, Problem 45PPSE

Solution

To state:

If the given statement is true for the given system of linear equations always, sometimes, or never.

The given statement is sometimes true.

Given:

The system of linear equations:

  $\left\{\begin{array}{l}y=x+3\\ y=mx+b\end{array}\right.$

The statement:

If $b=3$ , the system has exactly one solution.

Concepts Used:

If the equation of a line is expressed as $y=Mx+C$ , then $M$ represents the slope of the line and $C$ represents the $y$ intercept of the line.

If two lines have different slopes then the solutions of the system of these two lines has only one solution.

If two lines have equal slopes and their $y$ intercepts are not equal then the system of these two lines have no solution.

If two lines have equal slopes and their $y$ intercepts are equal then the system of these two lines have infinitely many solutions.

Calculations:

If $b=3$ then the given system of linear equations becomes:

  $\left\{\begin{array}{l}y=x+3\\ y=mx+3\end{array}\right.$

Comparing the lines with $y=Mx+C$ . It follows that the $y$ intercepts of both lines are equal to $3$ . The first line has a slope of $1$ while that of the second line is $m$ .

Two cases are possible.

Case I: if the slopes are equal $m=1$ .

The lines have both the slopes and the intercepts equal. Thus, they have infinitely many solutions.

Case II: if the slopes are not equal $m\ne 1$ .

The lines have different slopes. Thus, they have exactly one solution.

Conclusion:

There is one case $m=1$ for which the system of equations has infinitely many solutions and another case $m\ne 1$ for which the system of equations have exactly one solution.

Thus, the given statement that “If $b=3$ , the system has exactly one solution.” is sometimes true.