EXAMPLE 2 Solve a system with many solutions

Solve the system. Then classify the system as consistent and independent,consistent and dependent, or inconsistent.

4x – 3y = 8

8x – 6y = 16

Equation 1

Equation 2

SOLUTION

The graphs of the equations are the same line. So, each point on the line is a solution, and the system has infinitely many solutions. Therefore, the system is consistent and dependent.

EXAMPLE 3 Solve a system with no solution

Solve the system. Then classify the system as consistent and independent,consistent and dependent, or inconsistent.

2x + y = 4

2x + y = 1

Equation 1

Equation 2

SOLUTION

The graphs of the equations are two parallel lines. Because the two lines have no point of intersection, the system has no solution. Therefore, the system is inconsistent.

EXAMPLE 4 Standardized Test Practice

SOLUTION

Equation 1 (Option A)

y = 1 x + 30

Equation 2 (Option B)

EXAMPLE 4 Standardized Test Practice

y = x2.5

To solve the system, graph the equations y = x + 30 and y = 2.5x, as shown at the right.

EXAMPLE 4 Standardized Test Practice

Notice that you need to graph the equations only in the first quadrant because only nonnegative values of x and y make sense in this situation.

The lines appear to intersect at about the point (20, 50). You can check this algebraically as follows.

Equation 1 checks.

Equation 2 checks.

50 = 20 + 30

50 = 2.5(20)

ANSWERThe total costs are equal after 20 rides.

The correct answer is B.

GUIDED PRACTICE for Examples 2,3, and 4

Solve the system. Then classify the system as consistent and independent, consistent and dependent, or inconsistent.

4. 2x + 5y = 64x + 10y = 12

ANSWER Infinitely many solutions; consistent and dependent

GUIDED PRACTICE for Examples 2,3, and 4

Solve the system. Then classify the system as consistent and independent, consistent and dependent, or inconsistent.

5. 3x – 2y = 103x – 2y = 2

ANSWER no solution; inconsistent

GUIDED PRACTICE for Examples 2,3, and 4

Solve the system. Then classify the system as consistent and independent, consistent and dependent, or inconsistent.

6. –2x + y = 5y = –x + 2

ANSWER (–1, 3); consistent and independent

GUIDED PRACTICE for Examples 2,3, and 4

Solve the system. Then classify the system as consistent and independent, consistent and dependent, or inconsistent.

7. WHAT IF? In Example 4, suppose the cost of the monthly pass is increased to $36. How does this affect the solution?

ANSWER The number of rides increases to 24.