SOLUTION

EXAMPLE 1 A linear system with no solution

Show that the linear system has no solution.

3x + 2y = 10 Equation 1

3x + 2y = 2 Equation 2

Graph the linear system.

METHOD 1 Graphing

EXAMPLE 1 A linear system with no solution

ANSWER

The lines are parallel because they have the same slope but different y-intercepts. Parallel lines do not intersect, so the system has no solution.

EXAMPLE 1 A linear system with no solution

ANSWER

The variables are eliminated and you are left with a false statement regardless of the values of x and y. This tells you that the system has no solution.

Subtract the equations.

METHOD 2 Elimination

3x + 2y = 10

3x + 2y = 2

0 = 8 This is a false statement.

EXAMPLE 2 A linear system with infinitely many solutions

Show that the linear system has infinitely many solutions.

x – 2y = –4 Equation 1

Equation 2y = x + 212

SOLUTION

GraphingMETHOD 1

Graph the linear system.

EXAMPLE 2 A linear system with infinitely many solutions

ANSWER

The equations represent the same line, so any point on the line is a solution. So, the linear system has infinitely many solutions.

EXAMPLE 2 A linear system with infinitely many solutions

Substitute x + 2 for y in Equation 1 and solve for x.12

x – 2y = –4 Write Equation 1.

2Substitute x + 2 for y.1x – 2 x + 2 =1

2–4

METHOD 2 Substitution

The variables are eliminated and you are left with a statement that is true regardless of the values of x and y. This tells you that the system has infinitely many solutions.

ANSWER

–4 = –4 Simplify.

GUIDED PRACTICE for Examples 1 and 2

1. 5x + 3y = 6

–5x – 3y = 3

Tell whether the linear system has no solution or infinitely many solutions. Explain.

ANSWER

No solution. Sample answer: When you solve the system you get 0 = 9, which is a false statement.

GUIDED PRACTICE for Examples 1 and 2

2. y = 2x – 4

–6x + 3y = –12

Tell whether the linear system has no solution or infinitely many solutions. Explain.

ANSWER

Infinitely many solutions. Sample answer: When you solve the system you get 12 = 12, which is a true statement.