How do you find the point intersection of ln x=-7x

Along highway 20, approximate halfway point is at intersection with Highway 5 (Total trip = 88 miles)

Along I-85, approximate halfway point is intersection with I-75 (Total trip = 106 miles)

Since the y-variable is isolated in both equations, set the two right sides of the equations equal
x^2-7x+4=-8x^2+56x-86
Combine all like terms on one side of the equal sign
9x^2-63x+90=0. factor out the 9 to get 9(x^2-7x+10)=0
Now solve the equation (x^2-7x+10)=0. The solutions are x=5 and x=2
Calculate the y-value tof be y=-6
The two solutions are (2,-6) and (5,-6)
Graphically, here is how the two curves intersect. One intersection point in the screen capture is (2,-6). You can read that the other intersection point is (5,-6)

Here you are trying to find the intersection point between two curves. Since you could have drawn more than 2 curves, the calculator gives you the opportunity to choose the first curve using the arrow Up/ arrow down to select.

Once the curves are chosen, the calculator wants to know where (in the world is Carmen Sandiego) is the intersection point. And no it is not asking you to give the position, just what are the leftmost (1st guess) and rightmost (2nd guess) limits of the interval where the intersection point is. You use the left arrow to go a bit to the left of the intersection point (which you see on the graph) and press ENTER. Then you do the same for the 2nd guess.

Here you are trying to find the intersection point between two curves. Since you could have drawn more than 2 curves, the calculator gives you the opportunity to choose the first curve using the arrow Up/ arrow down to select.

Once the curves are chosen, the calculator wants to know where (in the world is Carmen Sandiego) is the intersection point. And no it is not asking you to give the position, just what are the leftmost (1st guess) and rightmost (2nd guess) limits of the interval where the intersection point is. You use the left arrow to go a bit to the left of the intersection point (which you see on the graph) and press ENTER. Then you do the same for the 2nd guess.

That line intersects an infinite number of things at each point. Before you can ask where it intersects something, you must specify what it intersects.

The solve( function was not able to find the intersection point because it lies outside the graph range. The window dimensions are too narrow to contain that point. I suggest you restrict the graph to the quadrant where you suspect the solution to be (example first quadrant if appropriate) : take Xmin=0, Ymin =0 and choose Xmax and Ymax large enough to enclose the intersection point.

See image below

To find the intersection (read interSECTION) of two curves you are prompted once for an initial guess. If the calculator does not find the intersection point , or if there are more than one intersection points, you have to restart the procedure.
For your information, the interCEPT (or y-intercept) is the ordinate of the point where a curve cuts the y-axis. It is easy to mistake one word for the other. They are not synonymous.

Hi,
You should check your understanding of what a function is. You are drawing two functions the ranges of which do not overlap, since one branch is positive and the other is negative. You know that the only two points where there could be overlapping are the points where y=0 for both functions. Why would you need the calculator to confirm to you what you already know.
To define the two branches you had to take the square root of some expression say y= SQRT(5-x^2). That is a circle centered on O(0,0) with radius SQRT(5). The two points where the positive branch intersects the negative one are for y=0, meaning x1= SQRT(5) or x2= -SQRT(5).
What do you think is the exact value of SQRT(5): 2.236067977....? No, because SQRT(5) is an irrational number that has an infinite number of digits and no matter how many additional digits you may align to determine it will not make that representation the EXACT value of SQRT(5).

That does not mean you will never be able to find an intersection of the two curves. Maybe, if you take y=SQRT(4-x^ 2) the calculator will be able to find the intersections but that will remain one case.In general the calculator will not find the intersection.

I hope that I convinced that it is futile to seek, with the help of the calculator, the intersection of two irrational functions ( for they are irrational not rational as you claim) that share only two points.

Hope it helps.

Hello,
The function Intersect from the CALCULATE menu finf the coordinates of a point at which two or more curves intersect.
To use it:
1. Draw the functions.

2. Press [2nd][CALC][5:Intersect]



The cursor is on one of the curves. Read the equation top of the screen. If it is one of the curves you want press [ENTER]. The cursor jumps to another curve (in this case the only other curve).

Read the equation on top of the screen to verify thst it is the correct one. Press [ENTER]. The calculator asks asks for a guess of the coordinates of the intersection point.


As the intersection point is to the left of the current cursor position, use the left arrow to move cursor closer to the point.


Press [ENTER], and wait for the solution. Here it is.


In your question you talk about y intercept. If you want to calculate the ordinate of the point where a curve intersects the Y-axis, it is more efficient to use the [2nd][CAL][1:Value] selection.

You enter X=0 and press [ENTER]. The cursor jumps on the first curve (Y1=) an gives you the y-intercept.

Notice the position of cursor on graph. The y-value at the bottom is its ordinate.
To get the y-intercept of the second curve, leave the cursor on y axis and press the DownArrow. Cursor jumps to tthe second curve.
Since the X=0 is still stored, the value of y is displayed directly.



Hope it helps.