Solving Systems of Equations By Graphing (6-1) Objective: Determine the number of solutions a system of linear equations has, if any. Solve a linear system of equations by the graphing method.

Possible Number of Solutions A system of equations is two or more equations. The ordered pair (x, y) that is a solution of both equations is the solution of the system. A system of two linear equations can have one solution, an infinite number of solutions, or no solution. –If a system has at least one solution, it is said to be consistent. The graphs intersect at one point, or are the same line. –If a consistent system has exactly one solution, it is said to be independent. If it has an infinite number of solutions, it is dependent. This means that there are unlimited solutions that satisfy both equations. –If a system has no solution, it is said to be inconsistent. The graphs are parallel.

Concept Summary Number of Solutions Exactly oneInfiniteNo solution Terminology Consistent and independent Consistent and dependent Inconsistent Graph

Example 1 Use the graph at the right to determine whether each system is consistent or inconsistent and if it is independent or dependent. a.y = -x + 1 y = -x + 4 Inconsistent

Example 1 Use the graph at the right to determine whether each system is consistent or inconsistent and if it is independent or dependent. b.y = x – 3 y = -x + 1 Consistent and Independent

Check Your Progress Choose the best answer for the following. A.Use the graph to determine whether the system is consistent or inconsistent and if it is independent or dependent. 2y + 3x = 6 y = x – 1 A.Consistent and independent B.Inconsistent C.Consistent and dependent D.Cannot be determine.

Check Your Progress Choose the best answer for the following. B.Use the graph to determine whether the system is consistent or inconsistent and if it is independent or dependent. y = x + 4 y = x – 1 A.Consistent and independent B.Inconsistent C.Consistent and dependent D.Cannot be determine.

Solve by Graphing One method of solving a system of equations is to graph the equations carefully on the same coordinate grid and find their point of intersection. This point is the solution of the system.

Example 2 Graph each system and determine the number of solutions that it has. If it has one solution, name it. a.y = 2x + 3 8x – 4y = -12 8x = -12 x = y = -12 y = 3 Same Line Infinitely many solutions

Example 2 Graph each system and determine the number of solutions that it has. If it has one solution, name it. b.x – 2y = 4 x – 2y = -2 x = -2 -2y = -2 y = 1 Parallel Lines No Solution x = 4 -2y = 4 y = -2

Example 2 Graph each system and determine the number of solutions that it has. If it has one solution, name it. c.y = x – 2 3y + 2x = 9 3y = 9 y = 3 2x = 9 x = 4.5 One Solution (3, 1)

Check Your Progress Choose the best answer for the following. A.Graph the system of equations. Then determine whether they system has no solution, one solution, or infinitely many solutions. If the system has one solution, name it. y = 2x + 3 y = ½ x + 3 A.One; (0, 3) B.No Solution C.Infinitely many D.One; (3, 3)

Check Your Progress Choose the best answer for the following. B.Graph the system of equations. Then determine whether they system has no solution, one solution, or infinitely many solutions. If the system has one solution, name it. x + 3y = 4 1 / 3 x + y = 0 A.One; (0, 0) B.No Solution C.Infinitely many D.One; (1, 3) x = 4 3y = 4 y = 1.3 y = - 1 / 3 x

Solving Systems by Graphing We can use what we know about systems of equations to solve many real-world problems that involve two or more different functions.

Example 3 Naresh rode 20 miles last week and plans to ride 35 miles per week. Diego rode 50 miles last week and plans to ride 25 miles per week. Predict the week in which Naresh and Diego will have ridden the same number of miles. –Equation for Naresh: y = 35x + 20 –Equation for Diego: y = 25x + 50  (3, 125) In 3 weeks they will both have ridden 125 miles.

Check Your Progress Choose the best answer for the following. –Alex and Amber are both saving money for a summer vacation. Alex has already saved $100 and plans to save $25 per week until the trip. Amber has $75 and plans to save $30 per week. In how many weeks will Alex and Amber have the same amount of money? A.225 weeks B.7 weeks C.5 weeks D.20 weeks y = 25x y = 30x + 75