Lesson 7-1

Graphing Systems of Equations

Definition• Systems of Equations- Two equations together. A

solution to a system of equations has 0, 1 or an infinite number of solutions.

• Consistent - If the graphs intersect or coincide, the system of equations is said to be consistent.

• Inconsistent - If the graphs are parallel, the systems of equation is said to be inconsistent.

• Consistent equations are independent or dependent. Equations with exactly one solution is independent. Equations with infinite solutions is dependent.

O x

y

O x

y

O x

y

Intersecting Lines Same Line Parallel Lines

Exactly 1 solution Infinitely Many No solutions

Consistent and independent

Consistent and dependent Inconsistent

Use the graph to determine whether each system has no solution, one solution, or infinitely many solutions.

O x

y y = x - 3

y = -x + 5

2x + 2y = -8

y = -x -4

2x + 2y = -8

Ex. 1

A. y = -x + 5y = x -3

B. y = -x + 5 2x + 2y = -8C. 2x + 2y = -8 y = -x - 4

Use the graph to determine whether each system has no solution, one solution, or infinitely many solutions.

O x

y

3x-3y=9

y = -x + 4

y = -x + 1

x - y = 3

a. y = -x + 1

y = -x + 4

b. 3x - 3y = 9

y = -x + 1

c. x - y = 3

3x - 3y = 9

Ex. 2

Graph each system of equations. Then determine whether the system has no solution, one solution, or infinitely many solutions. If the system has one solution, name it.

A. y = -x + 8 y = 4x -7

o

y

B. x + 2y = 5 2x + 4y = 2

Graph each system of equations. Then determine whether the system has no solution, one solution, or infinitely many solutions. If the system has one solution, name it.

A. 2x - y = -3 8x - 4y = -12

o

y

B. x - 2y = 4 x - 2y = -2

If Guy Delage can swim 3 miles per hour for an extended period of time and the raft drifts about 1 mile per hour, how may hours did he swim each day if he traveled an average of 44 miles per day?

Ex. 3 Write and Solve a System of Equations

Let s = the number of hours he swam and let f = the number of hours he floated each day.

The number of hours swimming

plus the number of hours floating equals hours in a day

s + f = 24

The daily miles traveled plus

the daily miles traveled floating equals

total miles in a day

3s + 1f = 44

s + f = 243s + f = 44

o

f

s

Guy spent about 10 hours swimming each day.

(10, 14)

BICYCLING: Tyler and Pearl went on a 20-kilometer bike ride that lasted 3 hours. Because there were many steep hills on the bike ride, they had to walk for most of the trip. Their walking speed was 4 kilometers per hour. Their riding speed was 12 kilometers per hour. How much time did they spend walking? (Hint: let r = number of hours riding and w = number of hours walking.)

r + w = 312 r + 4w = 20

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are needed to see this picture.

w

rThey walked for 2 hours