Chapter 6.1, Problem 15P

Reasoning Without graphing, decide whether the following system of linear equations has one solution, infinitely many solutions or no solution. Explain.

$\begin{array}{l}y=3x-5\hfill \\ 6x=2y+10\hfill \end{array}$

To determine

To Explain: Whether the system of linear equations has one solution, infinitely many solutions or no solution without graphing.

Answer to Problem 15P

The system of equations has infinitely many solutions. Because, overlapping lines have infinitely many solutions.

Explanation of Solution

Given information: $y=3x-5$ and $6x=2y+10$

Formula used: We use three conditions

i)If slope and y intercept of both equations are same then they have infinitely many solutions.

ii) If slope of both equations are same but the y intercepts are not same then they have NO solutions.

iii) If the slope of both equations are not same then they are intersecting lines and they have one solution.

Calculation:

First equation is $y=3x-5$ .

Solve the second equation for y.

  $6x=2y+10$

Subtract both sides $2y$

  $6x-2y=10$

Subtract both sides $6x$

  $-2y=-6x+10$

Divide both sides of the equation by -2.

  $y=3x-5$

So, both equations represent same line. So, lines are overlapping lines.

The overlapping lines have infinitely many solutions.

Hence the system of equations has infinitely many solutions.