7.5

Warm UpWarm Up

Lesson QuizLesson Quiz

Lesson PresentationLesson Presentation

Solve Special Types of Linear Systems

7.5 Warm-Up

1. Solve the linear system.2x + 3y = –9 x – 2y = 6

2. You buy 8 pencils for $8 at the bookstore. Standardpencils cost $0.85 and specialty pencils cost $1.25.How many specialty pencils did you buy?

ANSWER 3 specialty pencils

ANSWER (0, –3)

7.5 Example 1

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

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.

7.5 Example 1

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.

7.5 Example 2

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.

ANSWER

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

7.5 Example 2

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.

7.5 Guided Practice

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.

2. y = 2x – 4–6x + 3y = –12

ANSWER

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

7.5 Example 3

Without solving the linear system, tell whether the linear system has one solution, no solution, or infinitely many solutions.

a. 5x + y = –2 Equation 1

–10x – 2y = 4 Equation 2

SOLUTION

y = –5x – 2Write Equation 1 in slope-intercept form.

y = –5x – 2 Write Equation 2 in slope-intercept form.

ANSWER

Because the lines have the same slope and the same y-intercept, the system has infinitely many solutions.

7.5 Example 3

SOLUTION

Write Equation 1 in slope-intercept form.

Write Equation 2 in slope-intercept form.

ANSWER

Because the lines have the same slope but different y-intercepts, the system has no solution.

b. 6x + 2y = 3 Equation 1

6x + 2y = –5 Equation 2

y = –3x + 32

y = –3x – 52

7.5 Guided Practice

3. Without solving the linear system, tell whether it has one solution, no solution, or infinitely many solutions.

x – 3y = –15

2x – 3y = –18

Equation 1

Equation 2

ANSWERone solution

7.5 Example 4

ART

An artist wants to sell prints of her paintings. She orders a set of prints for each of two of her paintings. Each set contains regular prints and glossy prints, as shown in the table. Find the cost of one glossy print.

7.5 Example 4

SOLUTION

STEP 1

Write a linear system. Let x be the cost (in dollars) of a regular print, and let y be the cost (in dollars) of a glossy print.

45x + 30y = 465 Cost of prints for one painting

Cost of prints for other painting15x + 10y = 155

7.5 Example 4

STEP 2

Solve the linear system using elimination.

45x + 30y = 465

15x + 10y = 155

45x + 30y = 465

–45x – 30y = –4650 = 0

ANSWER

There are infinitely many solutions, so you cannot determine the cost of one glossy print. You need more information.

7.5 Guided Practice

4. WHAT IF? In Example 4, suppose a glossy print costs $3 more than a regular print. Find the cost of a glossy print.

ANSWER

A glossy print costs $8.

7.5 Lesson Quiz

Without solving the linear system, tell whether the linear system has one solution, no solution, or infinitely many solutions.

1. 4x + 2y = 12 y = –2x + 8

ANSWER no solution

ANSWER infinitely many solutions

2. –2x + 5y = 5

y = x + 125

7.5 Lesson Quiz

A group of 12 students and 3 teachers pays $57 for admission to a primate research center. Another group of 14 students and 4 teachers pays $69. Find the cost of one student ticket.

3.

ANSWER $3.50