A system of equations is a set of equations with the same unknowns.

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1 Unit B, Lesson.3 Solving Systems of Equations Functions can be used to model real-world situations where you may be interested in a certain event or circumstance while you are also concerned with another. Those two events may be described by two different variables or functions, each describing the very same thing. For instance, you may need to determine the number of tables to rent for a wedding reception, and your table choices include round tables that seat 8 people and square tables that seat 6 people. Or, you might want to compare two separate savings accounts that have different interest rates. Only by studying both models at the same time (that is, by solving the system) can the situation be fully understood, and the information you need extracted from it. In this lesson, you will apply technology to create tables and graphs that can reveal useful information about specific function models. A system of equations is a set of equations with the same unknowns. The solution set for a system of equations is the set of ordered pairs that represent all of the solutions to an equation or a system of equations. The solutions to a system of equations can be identified in a table by finding points at which the y-value of each function is the same for a particular x-value. In most technology-based tables, this means the y-values will be in the same row. Solutions to a system of equations can be identified in a graph by finding the intersection of the functions. The points of intersection of a graphed system of equations are the ordered pairs where the graphed functions intersect on a coordinate plane. The solution of a system of f ( x ) and g ( x ) occurs where f ( x ) g( x ) and the difference between these two values is 0. Also recall that the solution to a system of equations is written using braces, {}. Recall that a system of equations may have an infinite number of solutions, a finite number of solutions, or no real solutions. It is possible that a system of equations does not intersect. The solution to such a system of equations is the empty set, written as Ø or { } (a set of braces with no coordinates inside). A system of equations may intersect at every point, meaning the equations overlap completely. This type of system is known as a dependent system. A graphing calculator is often used to graph the equations within a system. You can approximate where the equations intersect from a table of values or use the trace feature of the calculator to find intersection points. To confirm the solution to a system, substitute the coordinates for x and y in both of the original equations. If the intersection point is approximated, the results will be nearly equal; otherwise, the results will be equal. UnitBLesson.3 1 3/9/018

2 Unit B, Lesson.3 Solving Systems of Equations (continued) Common Errors/Misconceptions not recognizing that the values in a table are subject to rounding errors that may be imposed elsewhere, and lead to the illusion of a solution not realizing that on a graphing calculator, the scale of the graph may be such that the pixilated character of a function may appear to be an intersection, when it is not not understanding that solutions must satisfy every function in a system Example 1 f ( x ) x Use a graph to solve the system. g( x ) 4 0.5x Verify that the identified coordinate pairs are solutions. 1. Create a graph of the two functions, f ( x ) and g ( x ). Graph each equation on the same coordinate plane, by hand or using a graphing calculator. To graph the system on your graphing calculator, use the directions appropriate to your model. On a TI-83/84: Step 1: Press the [Y=] button. Step : Type the first function into Y 1. Use [ND][x ] to enter radicals and the [X, T, θ, n] button for the variable x. Press [ENTER]. Step 3: Type the second function into Y. Press [ENTER]. Step 4: Press [ZOOM]. Arrow down to 6: Zstandard. Press [ENTER]. This method should yield a graph resembling the graph to the right. UnitBLesson.3 3/9/018

3 Unit B, Lesson.3 Solving Systems of Equations (continued) Example 1 (continued). Find the coordinates of any apparent intersections. Looking at the graph, estimate where the two equations intersect. It may be necessary to plot additional points. You may also determine the approximate point(s) of intersection on your graphing calculator: On a TI-83/84: Step 1: Press [ND][TRACE] to call up the CALC menu. Arrow down to 5: intersect. Press [ENTER]. Step : At the prompt, use the up/down arrow keys to select the Y 1 equation, and press [ENTER]. Step 3: At the prompt, use the up/down arrow keys to select the Y equation, and press [ENTER]. Step 4: At the prompt, use the arrow keys to move the cursor close to an apparent intersection, and press [ENTER]. The coordinates of the intersection points are displayed. The point of intersection is (4, ). 3. Verify that the identified coordinate pair is a solution to the original system of equations. In order for a coordinate to be a solution to a system, the coordinate must satisfy both equations of the system. Substitute the coordinate pair found in step into each of the original equations to determine whether it results in a true statement. The potential solution is (4, ). Let x 4, and let f ( x ) and g( x ). f ( x ) x 4 Substitute 4 for x and for f ( x ) g( x ) 4 0.5x 4 0.5( 4) Substitute 4 for x and for g ( x ) 4 The identified point (4, ) satisfies both f ( x ) and g ( x ). f ( x ) The solution to the system of equations g( x ) x 4 0.5x the intersection of the functions f ( x ) and g ( x ), and verified algebraically. is {(4, )}, as identified on the graph as UnitBLesson.3 3 3/9/018

4 Unit B, Lesson.3 Solving Systems of Equations (continued) Example Create a table to approximate the real solution(s), if any, to the system 1. Create a table of values for the two functions, f ( x ) and g ( x ). 1 f ( x ) 1 x. g( x ) x 1 Use a graphing calculator to generate a table of values for each function. On a TI-83/84: Step 1: Press the [Y=] button. Step : Type the first function into Y 1, using parentheses for the fraction and the [X, T, θ, n] button for the variable x. Press [ENTER]. Step 3: Type the second function into Y. Press [ENTER]. Step 4: Press [ND][WINDOW] to call up the TABLE SETUP screen. Set TblStart to 5 by typing [( ) ][5], and Tbl to 1, then press [ND][GRAPH] to call up the TABLE screen. A table of values for both equations will be displayed. The calculator will yield the table of values shown as follows. Use the up and down arrow keys as necessary to see the y-values for each x-value shown as follows.. Estimate the points of intersection. Compare the y-values for the same x-value in your completed table. Look for places where the difference between the values for f ( x ) and g ( x ) is decreasing. If the difference between f ( x ) and g ( x ) is increasing, then any intersection occurs in the other direction. The point of intersection occurs when f ( x ) is equal to g ( x ) or the difference is 0. Notice that f ( x ) equals g ( x ) when x 1 and when x 1. The points ( 1,0) and ( 1,) appear to be the solutions to the system. 3. Verify that the identified coordinate pairs are solutions to the original system of equations. Substitute the coordinates found in the previous step into each of the original equations to determine whether each results in a true statement. Start with the potential solution ( 1,0). Let x 1, and let f ( x ) 0 and g( x ) 0. 1 f ( x ) 1 x Substitute 1 for x and 0 for f ( x ) g( x ) x Substitute 1 for x and 0 for g ( x ) 0 0 UnitBLesson.3 4 3/9/018

5 Unit B, Lesson.3 Solving Systems of Equations (continued) Example (continued) The identified point ( 1,0) satisfies both f ( x ) and g ( x ), and is therefore a valid solution. Test the second potential solution ( 1,). Let x 1, and let f ( x ) and g( x ). 1 f ( x ) 1 x Substitute 1 for x and for f ( x ) 11 g( x ) x 1 11 Substitute 1 for x and for g ( x ) The identified point ( 1,) also satisfies both f ( x ) and g ( x ), and is therefore a valid solution. 1 f ( x ) 1 The solution to the system of equations x is ( 1,0),(1,), as identified using g( x ) x 1 a table of values showing where f ( x ) g( x ), and verified algebraically. Example 3 A system contains the equations y x 3 and 4x 4y 19. Solve the system by creating a table of values on a graphing calculator. 1. Determine whether the given equations are written in y = form and rewrite if necessary. Entering equations into calculators and spreadsheets generally demands that each equation be in y = form. While the first equation is already in y = form, the second is not. Use algebraic techniques to rewrite the equation 4x 4y 19 in y = form. 4x 4y 19 4y 4x 19 Subtract 4x from both sides y x y x 4 y x 4.75 Divide all terms by 4 The equation 4x 4y 19 written in y = form is y x Create a table of values for the two functions. Use the calculator steps shown in Example to create to a table of values for each equation of the defined system. UnitBLesson.3 5 3/9/018

6 Unit B, Lesson.3 Solving Systems of Equations (continued) Example 3 (continued) A table of values for both equations appears below. Let y 1 represent the equation y x 3 and y represent the equation y x Scrolling up and down through the calculator s list reveals that there are no x-values for which both functions have the same y-value. A closer inspection shows that the y-values for the two functions are closest in value when x 9 : y and y To better approximate the points of intersection, you must find additional values. Use the following directions to determine additional values on your graphing calculator. On a TI-83/84: Step 1: Press [ND][WINDOW] to call up the TABLE SETUP screen. Set TblStart to 8 and Tbl to 0.1, then press [ND][GRAPH] to call up the TABLE screen. Examine the revised table of values. The small changes in decimals can be hard to analyze. However, you might notice that when x 9., y y1. That relationship switches for x 9. 3, when y y1. This is an indication that something happens when 9. x In order to look at the table more closely, change the table settings again. Follow the directions previously outlined, but this time, start the table at 9., and set the interval to Examine the revised table again. When x 9. 5, both y 1 and y There is a potential solution to the system at ( 9.5, 4.5 ). 3. Continue examining the table of values for additional matching y-values. On your graphing calculator, set the table interval back to 0.1, or even 1, in order to verify there are no other potential solutions. For this system, it appears that the only potential solution is at ( 9.5, 4.5 ). UnitBLesson.3 6 3/9/018

7 Unit B, Lesson.3 Solving Systems of Equations (continued) Example 3 (continued) 4. Verify that the identified coordinate pair is a solution to the original system of equations. Substitute the coordinate pair into each of the original equations to determine whether it results in a true statement. In order for a coordinate pair to be a solution to a system, the coordinates must satisfy both equations of the system. Because your coordinate pair is estimated, it is quite possible that your results will be nearly equal, not exactly equal. The potential solution is ( 9.5, 4.5 ). Let x 9. 5, and let y y x Substitute 9.5 for x and 4.5 for y x 4y 19 4(9.5 ) 4( 4.5) 19 Substitute 9.5 for x and 4.5 for y The identified point ( 9.5, 4.5 ) satisfies both of the original equations. The solution to the system of equations defined by y x 3 and 4x 4y 19 is ( 9.5, 4.5), as identified using a table of values showing where y1 y, and verified algebraically. Example 4 Solve the system of equations f ( x ) 0.5x g( x ) 1 x 1. Graph both equations on the same set of axes. by graphing. Graph each equation on the same coordinate plane, by hand or using a graphing calculator. To graph the system on your graphing calculator, use the directions outlined in Example 1. The method should yield a graph resembling the graph to the right. UnitBLesson.3 7 3/9/018

8 Unit B, Lesson.3 Solving Systems of Equations (continued) Example 4 (continued). Find the coordinates of any apparent intersections. On your graph, estimate where the two equations intersect. It may be necessary to plot additional points or to adjust the window settings on your calculator. Looking at the graph indicates that these two functions may intersect twice, at approximately x 0 and x 4. One of those intersections appears to be the origin. You can more precisely determine the approximate point(s) of intersection using features on your graphing calculator. On a TI-83/84: Step 1: Press [ND][TRACE] to call up the CALC menu. Arrow down to 5: intersect. Press [ENTER]. Step : At the prompt, use the up/down arrow keys to select the Y 1 equation, and press [ENTER]. Step 3: At the prompt, use the up/down arrow keys to select the Y equation, and press [ENTER]. Step 4: At the prompt, use the arrow keys to move the cursor close to an apparent intersection, and press [ENTER]. The coordinates of the intersection points are displayed. The points of intersection are ( 0, 0) and ( 4, ). 3. Verify that the identified coordinate pairs are solutions to the original system of equations. Substitute each coordinate pair into the equations to determine whether each results in a true statement. The first potential solution is ( 0, 0). Let x 0, and let f ( x ) 0 and g( x ) 0. f ( x ) 0.5x 0 0.5(0) Substitute 0 for x and 0 for f ( x ) 0 0 g( x ) 1 x 0 1 Substitute 0 for x and 0 for g ( x ) The identified point ( 0, 0) satisfies both of the original equations, and is therefore a valid solution. Follow the same process to verify the potential solution ( 4, ). f ( x ) 0.5x 0.5( 4) Substitute 4 for x and for f ( x ) UnitBLesson.3 8 3/9/018

9 Unit B, Lesson.3 Solving Systems of Equations (continued) Example 4 (continued) g( x ) 1 x 1 Substitute 4 for x and for g ( x ) The identified point ( 4, ) also satisfies both of the original equations. The solution to the system of equations f ( x ) 0.5x and g( x ) 1 ( 0,0),( 4,) x as identified on the graph as the intersections of the functions f ( x ) and g ( x ), and verified algebraically. Example 5 Solve the system of equations 1 f ( x ) ( x 1) 3 g( x ) 4x 1x 1. Graph both equations on the same set of axes. 1x 3 is by graphing. Graph each equation on the same coordinate plane, by hand or using a graphing calculator. To graph the system on your graphing calculator, use the directions outlined in Example 1. The method should yield a graph resembling the graph to the right. UnitBLesson.3 9 3/9/018

10 Unit B, Lesson.3 Solving Systems of Equations (continued) Example 5 (continued). Find the coordinates of any apparent intersections. The graph indicates that these two functions never intersect. They come close where 0 x 1, but zooming in on that area of the graph shows that they never cross. Follow the steps outlined in Example 1 to zoom in to the area in question. The graph should resemble the following: There is no intersection; therefore, there is no solution to this system. There is no real solution to the system of equations 1 f ( x ) ( x 1) 3 g( x ) 4x 1x 1x 3. UnitBLesson /9/018