Solving Linear Systems by Graphing AII, 2.0: Students solve systems of linear equations and inequalities (in two or three variables) by substitution, with graphs, or with matrices. LA, 6.0: Students demonstrate an understanding that linear systems are inconsistent (have no solutions), have exactly one solution, or have infinitely many solutions
Solving Linear Systems by Graphing Objectives Key Words Solve a system of linear equations in two variables by graphing System of two linear equations Solution of a system of two linear equations
Prerequisite Check:If you do not know, you need to let me know Identify the slope of an equation Solve linear equations
Prerequisite Check:If you do not know, you need to let me know Tell whether the point is a solution of the equation Graph the equations
Take 5 minutes to discuss with someone next to you. • Things you may want to ask: • When have you compared the price of two items? • Have you ever had to make a choice between to objects or situations? • How much money did you spend the last time you were on the bus? Why do we solve a system of equations by graphing? Can you think of when you will use it? We are going to visit the following site for practice: http://www.classzone.com/cz/books/algebra_2_cs/resources/applications/animations/html/alg207_ch03_pg155.html
Take a few moments to think about how many solutions we can possibly have given two random equations. • Things you may want to ask your partner: • Do we need to have a solution? • Is there always going to be a solution? • What if there is more than one solution? • When will we have no solutions? How many solutions can we have by graphing?
There are the graphs withOne Solution The graph of the system is a pair of lines that intersects in one point The lines have different slopes. The system has exactly one solution.
There are the graphs withMany Solutions The graph of the system is a pair of identical lines The lines have the same slope and the same y-intercept The system has infinitely many solutions
There are the graphs withNo Solutions The graph of the system is a pair of parallel lines, which do not intersect The lines have the same slope and different y-intercept The system has no solutions
Steps: • For each equation plot two points and draw the line: • Pick two inputs to get two outputs • For example, pick x=0 and x=1 • Find the intersections • You may approximate the answer • Verify your answer is correct. • Plug in you x and y value and check if it is true Solve a system by graphing
Example 1 = 3x – y 3 = x + 2y 8 SOLUTION Graph both equations, as shown. From the graph, you can see the lines appear to intersect at 2,3 ( ). Solve a System by Graphing Solve the system by graphing. Then check your solution algebraically. Equation 1 Equation 2
Example 1 ? ? = = Equation 1 Equation 2 ( 2, 3 ). = x + 8 2y ? 2 8 2 + 3 – 3 3 = The solution of the system is ? 6 – 3 3 6 = 2 8 + = 3x – y 3 = = 3 8 3 8 ANSWER ( ( ) ) 2 3 Solve a System by Graphing You can check the solution by substituting 2 for x and 3 for y into the original equations.
Checkpoint + = y 2 x 9 + = y – x 3 ANSWER – ( 2, 5 ) Solve a System by Graphing Solve the system by graphing. Then check your solution. 1.
Checkpoint – – + – = x y 1 = x 3y 1 ANSWER ( 1, 0 ) Solve a System by Graphing Solve the system by graphing. Then check your solution. 2.
Checkpoint – x + 4y 2 – = 2x 3y 6 ANSWER ( 6, 2 ) Solve a System by Graphing Solve the system by graphing. Then check your solution. 3. =
b. – 2x y 1 = x + 2y 4 = – – 4x + 2y 2 = SOLUTION x + 2y 1 = a. Because the graph of each equation is the same, each point on the line is a solution. So, the system has infinitely many solutions. Example 2 Systems with Many or No Solutions Tell how many solutions the linear system has. a.
Example 2 Systems with Many or No Solutions b. Because the graphs of the equations are two parallel lines, the two lines have no point of intersection. So, the system has no solution.
Days in San Diego Days in Anaheim Total Budget SOLUTION VERBAL MODEL Days in Anaheim Days in San Diego Total vacation time You can use a verbal model to write a system of linear equations. + = Daily cost in San Diego Daily cost in Anaheim + • + • = Example 3 Write and Use a Linear System Vacation You are planning a 7-day trip to California. You estimate that it will cost $300 per day in San Diego and $400 per day in Anaheim. Your total budget for the trip is $2400. How many days should you spend in each city?
Equation 2 (total budget) + x y 7 = + 300x 400y 2400 = Equation 1 (total vacation time) ALGEBRAIC MODEL Total budget 2400 = Example 3 Write and Use a Linear System (days) Days in San Diego x LABELS = (days) Days in Anaheimy = (days) Total vacation time 7 = (dollars per day) Daily cost in San Diego 300 = (dollars per day) Daily cost in Anaheim 400 = (dollars)
The lines appear to intersect at . ( ) 4,3 Example 3 Write and Use a Linear System Graph both equations only in the first quadrant because the only values that make sense in this situation are positive values of x and y.
Equation 1 + + x y 4 3 7 = = Equation 2 ( ) 4, 3 ( ) + + 3 300x 400y 300 400 2400 = = ANSWER The solution is . You should plan to spend 4 days in San Diego and 3 days in Anaheim. ( ) 4 Example 3 Write and Use a Linear System CHECK Substitute 4 for x and 3 for y in the original equations.
Checkpoint 6. 4. 2x + 3y 1 = 0 ANSWER 4x + 6y 3 = 1 ANSWER 5. – x 4y 5 = – – x + 4y 5 = infinitely many solutions ANSWER – x 5y 5 = x + 5y 5 = Write and Use Linear Systems Tell how many solutions the linear system has.
Checkpoint 7. Vacation Your family is planning a 6-day trip to Florida. You estimate that it will cost $450 per day in Tampa and $600 per day in Orlando. Your total budget is $3000. How many days should you spend in each city? ANSWER 4 days in Tampa and 2 days in Orlando Write and Use Linear Systems Tell how many solutions the linear system has.
Conclusions Summary Assignment • How do you solve a system of linear equations graphically? • Graph the linear equations and estimate the point where the graphs cross. Check the solution algebraically. Pg128 #(10,16,28,36,37) Due by the end of the class.