1. Solutions to Systems of Linear Equations. Determine whether the ordered pairs are solutions to the system. x y 6. 3x y 2

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1 78 Chapter Sstems of Linear Equations Section. Concepts. Solutions to Sstems of Linear Equations. Dependent and Inconsistent Sstems of Linear Equations. Solving Sstems of Linear Equations b Graphing Solving Sstems of Linear Equations b Graphing. Solutions to Sstems of Linear Equations A linear equation in two variables has an infinite number of solutions that form a line in a rectangular coordinate sstem. Two or more linear equations form a sstem of linear equations. For eample: 0 A solution to a sstem of linear equations is an ordered pair that is a solution to each individual linear equation. Eample Determining Solutions to a Sstem of Linear Equations Determine whether the ordered pairs are solutions to the sstem. a., b. 0, 6 a. Substitute the ordered pair, into both equations: True True Because the ordered pair, is a solution to both equations, it is a solution to the sstem of equations. b. Substitute the ordered pair 0, 6 into both equations: True False Because the ordered pair 0, 6 is not a solution to the second equation, it is not a solution to the sstem of equations Determine whether the ordered pairs are solutions to the sstem. 8 8 a. (, ) b. (, 0) a. No b. Yes

2 Section. Solving Sstems of Linear Equations b Graphing 79 A solution to a sstem of two linear equations ma be interpreted graphicall as a point of intersection between the two lines. Notice that the lines intersect at, (Figure -). (, ) 6 Figure -. Dependent and Inconsistent Sstems of Linear Equations When two lines are drawn in a rectangular coordinate sstem, three geometric relationships are possible:. Two lines ma intersect at eactl one point.. Two lines ma intersect at no point. This occurs if the lines are parallel.. Two lines ma intersect at infinitel man points along the line. This occurs if the equations represent the same line (the lines are coinciding). If a sstem of linear equations has one or more solutions, the sstem is said to be a consistent sstem. If a linear equation has no solution, it is said to be an inconsistent sstem. If two equations represent the same line, then all points along the line are solutions to the sstem of equations. In such a case, the sstem is characterized as a dependent sstem. An independent sstem is one in which the two equations represent different lines. Solutions to Sstems of Linear Equations in Two Variables One unique solution No solution Infinitel man solutions One point of intersection Parallel lines Coinciding lines Sstem is consistent. Sstem is inconsistent. Sstem is consistent. Sstem is independent. Sstem is independent. Sstem is dependent.

3 80 Chapter Sstems of Linear Equations. Solving Sstems of Linear Equations b Graphing Eample Solving a Sstem of Linear Equations b Graphing Solve the sstem b graphing both linear equations and finding the point(s) of intersection. 6 To graph each equation, write the equation in slope-intercept form m b. Slope: Slope: From their slope-intercept forms, we see that the lines have different slopes, indicating that the lines must intersect at eactl one point. Using the slope and -intercept we can graph the lines to find the point of intersection (Figure -). 6 Point of intersection (, ) Figure - The point, appears to be the point of intersection. This can be confirmed b substituting and into both equations.. (, ) 9 True True The solution is,.. Solve b using the graphing method.

4 Section. Solving Sstems of Linear Equations b Graphing 8 TIP: In Eample, the lines could also have been graphed b using the - and -intercepts or b using a table of points. However, the advantage of writing the equations in slope-intercept form is that we can compare the slopes and -intercepts of each line.. If the slopes differ, the lines are different and nonparallel and must cross in eactl one point.. If the slopes are the same and the -intercepts are different, the lines are parallel and do not intersect.. If the slopes are the same and the -intercepts are the same, the two equations represent the same line. Eample Solve the sstem b graphing. Solving a Sstem of Linear Equations b Graphing The first equation 8 can be written as. This is an equation of a vertical line. To graph the second equation, write the equation in slopeintercept form Figure - 8 (, 0) The graphs of the lines are shown in Figure -. The point of intersection is (, 0). This can be confirmed b substituting (, 0) into both equations. 8 8 True True The solution is (, 0).. Solve the sstem b graphing.. (, )

5 8 Chapter Sstems of Linear Equations Eample Solving a Sstem of Equations b Graphing Solve the sstem b graphing Figure - To graph the line, write each equation in slope-intercept form Because the lines have the same slope but different -intercepts, the are parallel (Figure -). Two parallel lines do not intersect, which implies that the sstem has no solution. The sstem is inconsistent.. Solve the sstem b graphing. 0 Eample Solve the sstem b graphing. Solving a Sstem of Linear Equations b Graphing 8. No solution; inconsistent sstem Write the first equation in slope-intercept form. The second equation is alread in slope-intercept form

6 Section. Solving Sstems of Linear Equations b Graphing 8 Notice that the slope-intercept forms of the two lines are identical. Therefore, the equations represent the same line (Figure -). The sstem is dependent, and the solution to the sstem of equations is the set of all points on the line. Because not all the ordered pairs in the solution set can be listed, we can write the solution in set-builder notation. Furthermore, the equations 8 and represent the same line. Therefore, the solution set ma be written as, 0 6 or. Solve the sstem b graphing. Figure -, Calculator Connections The solution to a sstem of equations can be found b using either a Trace feature or an Intersect feature on a graphing calculator to find the point of intersection between two curves. For eample, consider the sstem 6 First graph the equations together on the same viewing window. Recall that to enter the equations into the calculator, the equations must be written with the -variable isolated. That is, be sure to solve for first. Isolate. 6 6 B inspection of the graph, it appears that the solution is,. The Trace option on the calculator ma come close to, but ma not show the eact solution (Figure -6). However, an Intersect feature on a graphing calculator ma provide the eact solution (Figure -7). See our user s manual for further details.. {, 0 }; infinitel man solutions; dependent sstem