Rev Name Date. Most equations taught in algebra classes can and should be solved using algebra to get exact solutions.

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1 Name Date TI-84+ GC 3 Solving Equations Using x-intercept of Difference LHS RHS = (Method ) Objectives: Review: set an equation equal to, equation of horizontal line, x-axis, x-intercept, zero Understand the relationship between a solution and an x-intercept or zero Use the Zero calculation in the GC Recognize when the solution is exact or approximate Most equations taught in algebra classes can and should be solved using algebra to get exact solutions. Hard equations, like many real-world problems, often cannot be solved using algebraic methods. When an equation cannot be solving using algebra, it s useful to find an approximate solution using a calculator and graphs. We can use the GC to find a solution to an equation. If we have to round, the answer will be approximate. Different textbooks say this differently, but these instructions all mean the same thing: Solve graphically Solve numerically Use a graphing calculator to find a solution Approximate the solution Find an approximate solution The first method, intersection of graphs, is taught in the previous GC exercise. Here, we learn a second method, x-intercept of difference of graphs. To use the x-intercept or zero method, we will need to rearrange the equation so it equals. Example 1: Re-write 4x 6= 8x 18 so that the RHS is zero by subtracting from both sides. Subtract 8x from both sides: 4x 8x 6= 18 Add 18 to both sides: 4 x 8x 6+ 18= Answer: 4 x 8x 6+ 18= For the GC, it s optional to simplify: 4 x + 1= Answer: 4 x + 1= Both answers will give correct solutions when used in the GC. We now have a new equation LHS RHS =. If using Intersection of graphs, we d graph y1 = LHS RHS and y =. Recall: y= is a horizontal line. It s the same line as the x-axis. The point(s) where a graph crosses the x-axis are called x-intercepts or zeros. The GC calls it a zero. So the point of intersection (and location of the solution) is where the new expression LHS RHS crosses the x-axis, in other words, the x-intercept(s). IMPORTANT: The solution is the x-coordinate of the zero (not the y-coordinate). Copyright 11 by Martha Fidler Carey. Permission to reproduce is given only to current Southwestern College instructors and students.

2 TI-84+ GC 3 Solving Equations Using x-intercept of Difference LHS RHS = (Method ) p. NOTE: Don t use Trace to find the x-intercept. Trace may get close, but it often won t be right. The GC can calculate the x-intercept using Zero, in the CALC menu. To use a graphing calculator to find a solution (sometimes approximate) to an equation using the x- intercept of difference method: Step : Subtract one side to get an equation equal to zero: LHS RHS = Step 1: Graph y = LHS RHS 1. CAUTION: This method uses only one graph. Clear (or turn off) all other functions in the Y= menu. Step : Make sure you can see the x-intercept in the GC window. If not, adjust the window. Step 3: Use the Zero calculation, in the CALC menu, which is. This calculation has four steps, (which are different from those used by Intersect). Step 3a: Select option, zero, from the CALC menu. Step 3b: GC asks Left bound? Press or until the cursor is visible on the left side of the x-intercept. If the cursor is not visible, look at the bottom of the screen; the GC will tell you the coordinates of the cursor. Press appears on the screen, marking the left bound. to confirm the left bound. A small triangle Step 3c: GC asks Right bound? Press until the cursor passes to the right side of the x-intercept. Press the right bound. to confirm. Another small triangle appears on the screen, marking Step 3d: Lastly, the GC asks Guess?. Move the cursor closer to the x-intercept using the buttons. Your guess doesn t have to be perfect. Press. CAUTION: Zero is fussier than Intersect don t press ENTER three times or you ll get this error: Copyright 11 by Martha Fidler Carey. Permission to reproduce is given only to current Southwestern College instructors and students.

3 TI-84+ GC 3 Solving Equations Using x-intercept of Difference LHS RHS = (Method ) p.3 Example : Use the x-intercept of difference method to find the solution of 4x 6= 8x 18. Step : Set equation equal to. We did this in Example 1. 4 x 8x 6+ 18= Step 1: Graph y = 4x 8x Step : Make sure you can see the x-intercept in the GC window. If not, adjust the window. The x-intercept is visible on or near (3,). Step 3: Use Zero to find the x-intercept. Step 3a: Select option, zero, from the CALC menu. Step 3b: Move cursor to the left of x-intercept, and press ENTER to select Left bound? This screen tells us that the cursor is located at (,1). Press several times until the cursor is visible:, then press. Notice that a small triangle, pointing to the right, appears on the screen.. Copyright 11 by Martha Fidler Carey. Permission to reproduce is given only to current Southwestern College instructors and students.

4 TI-84+ GC 3 Solving Equations Using x-intercept of Difference LHS RHS = (Method ) p.4 Example, continued Step 3c: Move cursor to the right of x-intercept, and press ENTER to select Right bound? Press several times to move the cursor to the right of the x-intercept. Then press. Notice that a small triangle, pointing to the left, appears on the screen. The x-intercept must be between these two triangles, or the GC s calculation will fail. Step 3d: Move the cursor near x-intercept and press ENTER to select this Guess. Press once or twice to select a value that s close to the x-intercept. The Guess doesn t have to be very close, but it does have to be between the two triangles. Press. Answer: x=3 IMPORTANT: If the x-intercept is not visible in the GC window, the calculator won t find it. Adjust the window if necessary. Example 3: Use the x-intercept of difference method to find the solution of 7 x x 9 = 6 x+ 1 x+ 6. ( ) ( ) ( ) ( ) Step : Set equation equal to. (Simplifying is optional.) Add to both sides: ( 7 x + 13) ( 5x 9) + 6( x+ 1) = ( x+ 6) Add to both sides: ( 7 x + 13) ( 5x 9) + 6( x+ 1) + ( x+ 6) = Copyright 11 by Martha Fidler Carey. Permission to reproduce is given only to current Southwestern College instructors and students.

5 TI-84+ GC 3 Solving Equations Using x-intercept of Difference LHS RHS = (Method ) page 5 Example 3, continued Step 1: Graph = ( 7x+ 13) ( 5x 9) + 6( x+ 1) + ( x 6) y. 1 + Step : Make sure you can see the x-intercept in the GC window. If not, adjust the window. The x-intercept is not visible, and we have no graph to give clues to its location, so both increase Xmax and decrease Xmin. Step 3: Use Zero to find the x-intercept. The cursor is at (,18), press many times. Then press once or twice Answer: x = 1. Copyright 11 by Martha Fidler Carey. Permission to reproduce is given only to current Southwestern College instructors and students.

6 TI-84+ GC 3 Solving Equations Using x-intercept of Difference LHS RHS = (Method ) p.6 NOTE: If the graph does not cross the x-axis, there is no solution. One way this can happen is if the graph is a horizontal line (other than the x-axis). Example 4: Use the x-intercept of difference method to find the solution of 3 x 7= 3x+ If you are wide awake, you will notice right away that these two expressions are lines that have the same slope but different y-intercepts, making them parallel. Answer: No solution. But if you re still sleepy, set equal to zero. 3 x 7 3x = Hopefully, you ll notice that 3x-3x=, giving 7, a contradiction. Answer: No solution. But if you re sleepwalking, graph: This graph has no x-intercept. Answer: No solution. NOTE: If there are two (or more) points of intersection, use the same method twice (or more): once for each solution. You must choose the Guess? more carefully when there is more than one. Example 5: Use the x-intercept of difference method to find the solution of 9 x = 7. Round to the nearest hundredth. Step : Set equal to zero: 9 x 7= 1 Step 1: Graph y = 9 x 7. Step & 3: Notice two x-intercepts, and use zero twice. Press 5-6 times,, twice,. Copyright 11 by Martha Fidler Carey. Permission to reproduce is given only to current Southwestern College instructors and students.

7 TI-84+ GC 3 Solving Equations Using x-intercept of Difference LHS RHS = (Method ) p.7 Example 5, continued. about 1 times, about 5 times,. Answer: x 1.41, Example 6: If using the x-intercept of difference method to solve 3x 7= 3x 7, what graph will you get? What does that graph mean about the solution to this equation? Set equal to zero: 3 x 7 3x+ 7= (Simplifying helps! =.) Graph y 1 =. or y = 3x 7 3x 7 : 1 + The graph is the same as the x-axis. There are infinitely many solutions. Answer: all real numbers Copyright 11 by Martha Fidler Carey. Permission to reproduce is given only to current Southwestern College instructors and students.

8 TI-84+ GC 3 Solving Equations Using x-intercept of Difference LHS RHS = (Method ) p.8 Practice. 1) Describe what the graphs of these equations all have in common: y = 3, y = 5, y = 17, y =. ) What is the equation of a horizontal line passing through the point (,)? 3) The graph of y = has another name. What is it? 4) If we want to see y = in the GC, do we need to put it into the Y= menu? 5) What is a point where a graph intersects the x-axis called? 6) What is the y-coordinate of any x-intercept? 7) Solve the equation 5 ( x ) + 15= graphically using the x-intercept of difference method. 8) Solve the equation x + 6= x 4 graphically using the x-intercept of difference method. 9) Solve x π = 4 x algebraically. 1) Solve x π = 4 x graphically using the x-intercept of difference method. Round to the nearest hundredth. 11) Check that your solution to 9) is the same as your solution to 1). 1) Solve ( x ) ( x+ 14) = ( x 6) ( 1 x) 3 graphically using the x-intercept of difference method. 13) Solve ( ) ( 5x 9) = 6( x+ 1) ( x+ 6) method. x graphically using the x-intercept of difference = 14) Solve 5+x 8 graphically using the x-intercept of difference method. Round to the nearest hundredth. 15) Solve 5 x = 7 graphically using the x-intercept of difference method. Copyright 11 by Martha Fidler Carey. Permission to reproduce is given only to current Southwestern College instructors and students.

9 TI-84+ GC 3 Solving Equations Using x-intercept of Difference LHS RHS = (Method ) p.9 Solutions 1) These graphs are all horizontal lines. ) y = 3) The x-axis. 4) No. The x-axis is automatically included in any graph in the standard window. 5) An x-intercept 6) 7) 5 ( x ) + 15 = y = 5( x ) + 15 or y = 5( x ) several times Answer: x=3 8) x + x+ 6+ 4= or 1= Answer: no solution 4+ π 9) x π = 4 x x + x π = 4 3x = 4+ π, Answer: x = 3 1) x + x π 4= y 1 = 3x π 4 Answer: x π 11) Answer: x = Copyright 11 by Martha Fidler Carey. Permission to reproduce is given only to current Southwestern College instructors and students.

10 TI-84+ GC 3 Solving Equations Using x-intercept of Difference LHS RHS = (Method )p.1 1) ( 3 x ) ( x+ 14) + ( x 6) + ( 1 x) = y = ( 3x ) ( x+ 14) + ( x 6) + ( 1 x) 1 Don t know where the graph is; increase Xmax and decrease Xmin. Answer: x=-1 = 14) 5 8+ x y 1 = 3+ x calculate twice. Answer: x=15 13) ( 7 x + 13) ( 5x 9) + 6( x+ 1) + ( x+ 6) = y = ( 7x+ 13) ( 5x 9) + 6( x+ 1) + ( x 6) 1 + x 1.73, 1.73 Answer: = 15) 5 x = x y1 = x Graph has no x-intercepts. Answer: no solution Copyright 11 by Martha Fidler Carey. Permission to reproduce is given only to current Southwestern College instructors and students.

Introduction to the Graphing Calculator for the TI-86

Algebra 090 ~ Lecture Introduction to the Graphing Calculator for the TI-86 Copyright 1996 Sally J. Glover All Rights Reserved Grab your calculator and follow along. Note: BOLD FACE are used for calculator