( 3, 4) means x = 3 and y = 4
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1 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 = 2x + 3 y = 5x 4 Examples of a System of Linear Equations y = 3x 7 2x + y =17 y = 4 x y = 4 What is a solution to a system of Linear Equations? The solution to a system of linear equations is all the ordered pairs that make both equations true at the same time. If an ordered pair (x, y) is solution to the system then when you put the values for the x and y coordinates into BOTH equations then the ordered pair must make both equations true. If that x and y pairdo not make both equations true then the point is not a solution. Example 1 Example 2 Is ( 7, 3) a solution to 2x + y =17 x y = 4 ( 7, 3) means x = 7 and y = 3 Is ( 3, 4) a solution to x + 2y = 9 3x + y = 5 ( 3, 4) means x = 3 and y = 4 x = 7 and y = 3 2x + y =17 2(7) + (3) =17 x y = 4 7 (3) = 4 ( 7, 3) works in both equations so YES it is a solution. x = 2 and y = 4 x + 2y = 9 ( 3) + 2(4) =11 3x + y = 5 3( 3) + (4) = 5 ( 3, 4) does not work in Equation A so NO it is not a solution. Math 370 Section 11 2B Page 1 of Eitel
2 Example 3 Example 4 Is ( 6,4) a solution to Equation A 3x + 2y = 10 Equation B y = 2 3 x + 8 ( 6,4) means x = 6 and y = 4 x = 6 and y = 4 3x + 2y = Equation B y = 2 3 x + 8 ( ) + 2( 4) = 10 4 = 2 3 ( 6) + 8 ( 6,4) Does NOT work in Equation A so NO it is not a solution. Is 1 2,2 a solution to 3 2x 6y = 3 4x + 3y = 4 1 2,2 means x = and y = 2 3 x = 1 2 and y = 2 3 2x 6y = Equation A = 3 Equation B 4x + 3y = = 4 1 2,2 works in both equations 3 so YES it is a solution. Math 370 Section 11 2B Page 2 of Eitel
3 Solving a System of Linear Equations by Graphing In this chapter we will list two linear equations and ask you to graph each of them on the same graph. Each line will go through points that make itʼs equation true. We will be trying to find a point that makes both equations true at the same time. The solution to a system of linear equations is all the ordered pairs that make both equations true at the same time. y = x + 2 y = 2x 7 Any point on the line that represents Equation A is a solution to Equation A. Several points have been highlighted in red on the graph that are solutions to Equation A. 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 Equation B. As you can see, the points on Equation A are different then the points for Equation B except in one case. The point (3, 1) is the only point that is is on both lines. It is the only ordered pair that will make both equations true. We call the point (3, 1) the solution to the system of two lines. The point (3, 1) is the solution to the system of two lines. Check to see if (3, 1) is a solution. y = x + 2 y = 2x 7 1 = (3) = 2(3) 7 is true is true Math 370 Section 11 2B Page 3 of Eitel
4 Example 1 Solve the system of equations by graphing. List your answers as an ordered pair. Equation A y = 3 2 x + 2 Equation B y = 1 2 x 2 The lines have intersect at the point ( 2, 1). That point is on both lines and the x and y values will make both Equation A and Equation B true. Answer: ( 2, 1) Check: Equation A y = 3 2 x + 2 Equation B y = = 3 2 ( 2) + 2 True x 2 1 = ( ) 2 True Math 370 Section 11 2B Page 4 of Eitel
5 Example 2 Equation A y = 3 2 x +1 Equation B y = 1 2 x + 5 The lines intersect at the point (4, 7). That point is on both lines and the x and y values will make both Equation A and Equation B true. Answer: (4,7) Check: x + 2y =10 (4) + 2(7) =10 True y = 3 2 x +1 7 = 3 ( 2 4 ) +1 True Math 370 Section 11 2B Page 5 of Eitel
6 Do all systems intersect at a point and have one ordered pair as a solution? The graphs of the system of two lines can have three possible outcomes. Each of the different possible outcomes has a different format for the answer. Math 370 Section 11 2B Page 6 of Eitel
7 Examples of the three possible outcomes. Math 370 Section 11 2B Page 7 of Eitel
( 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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