( 7, 3) means x = 7 and y = 3. ( 7, 3) works in both equations so. Section 5 1: Solving a System of Linear Equations by Graphing

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1 Section 5 : Solving a Sstem of Linear Equations b Graphing What is a sstem of Linear Equations? A sstem of linear equations is a list of two or more linear equations that each represents the graph of a line. This section of the book will limit the sstems to just two linear equations Eamples of a Sstem of Linear Equations = + 3 = 5 = 3 7 = Equation C + =7 = 3 = 8 What is a solution to a sstem of Linear Equations? The solution to a sstem of linear equations is all the ordered pairs that make both equations true at the same time. If an ordered pair (, ) is solution to the sstem then when ou put the values for the and coordinates into BOTH equations then the ordered pair must make both equations true. If that and pair do not make both equations true then the point is not a solution. Eample Eample Is ( 7, 3) a solution to + =7 = ( 7, 3) means = 7 and = 3 Is ( 3, ) a solution to + = = 5 ( 3, ) means = 3 and = and to see if the work in both equations and to see if the work in both equations = 7 and = 3 + =7 (7) + (3) =7 = 7 (3) = ( 7, 3) works in both equations so YES it is a solution. = 3 and = + = = 5 3, ( 3) + () = 3( 3) + () = 5 ( ) DOES NOT work in so NO it is not a solution. Math Section 5 Page Eitel

2 Eample 3 Eample Is ( 6,) a solution to 3 + = = Is, a solution to 3 6 = = ( 6,) means = 6 and = and to see if the work in both equations = 6 and = 3 + = 3 6 = ( ) + ( ) = = 3 ( 6) + 8 ( 6,) Does NOT work in so NO it is not a solution., means = 3 and = 3 and to see if the work in both equations = and = 3 6 = = = =, works in both equations 3 so YES it is a solution. Math Section 5 Page Eitel

3 Solving a Sstem of Linear Equations b Graphing In this chapter we will list two linear equations and ask ou to graph each of them on the same graph. Each line will go through points that make itʼs equation true. We will be tring to find a point that makes both equations true at the same time. The solution to a sstem of linear equations is all the ordered pairs that make both equations true at the same time. = + = 7 An point on the line that represents is a solution to. Several points have been highlighted in red on the graph that are solutions to. The table for those points shows several of the infinite number of ordered pairs that make that equation true. The same has been done for the line that represents but the points are in black. As ou can see, the points on are different then the points for ecept in one case. The point at (3, ) is the onl point that is is on both lines. It is the onl ordered pair that will make both equations true. We call the point (3, ) the solution to the sstem of two equations. = + = 7 X Y X Y = = 7 Check to see if (3, ) is a solution. The point (3, ) is the solution to the sstem of two lines. = + = 7 = (3) + = (3) 7 is true is true Math Section 5 Page 3 Eitel

4 Eample Solve the sstem of equations b graphing. List our answers as an ordered pair. = 3 + = = 3 + = The lines have intersect at the point (, ). That point is on both lines and the and values will make both and true. Answer: (, ) Check: = 3 + = 3 ( ) + True = = ( ) True Math Section 5 Page Eitel

5 Eample Solve the sstem of equations b graphing. List our answers as an ordered pair. = = First Solve for + = = + = + 5 = 3 + = = = 3 + The lines intersect at the point (, 7). That point is on both lines and the and values will make both and true. Answer: (,7) Check: + = () + (7) = True = = 3 ( ) + True Math Section 5 Page 5 Eitel

6 Do all sstems intersect at a point and have one ordered pair as a solution? The graphs of the sstem of two lines can have three possible outcomes. Each of the different possible outcomes has a different format for the answer.. The lines have one point in common. (,) The lines intersect at one point. If two lines cross over each other then the will have eactl one point in common. The and coordinates of that point will make both equations true. That point is the solution to the sstem. Answer: (, ). The lines have no points in common. The lines are parallel If two lines have the same slope and different intercepts then the are parallel and the will have no points in common. There are no and coordinates that will make both equations true. Answer: No Solution 3. The lines coincide. When ou graph both lines the lie on top of one another. The two lines have the same equation and all the points on one line are also on the other line. All the points on the linear equation will make both equations true. Answer: All points on = m + b where the actual equation is written in place of the = m + b Math Section 5 Page 6 Eitel

7 Eamples of the three possible outcomes. (,). The lines have one point in common. = + = the lines cross at the point (,) Answer: (, ). The lines have no points in common. = + = 3 The lines are parallel and do not have an points in common. Answer: No Solution 3. The lines coincide. + = = when ou solve for ou get = = Equations A and B are the same line so all the points that make true make true also. Answer: All points on = Math Section 5 Page 7 Eitel

( 3, 4) means x = 3 and y = 4

( 3, 4) means x = 3 and y = 4 11 2B: Solving a System of Linear Equations by Graphing What is a system of Linear Equations? A system of linear equations is a list of two linear equations that each represents the graph of a line. y