How to get Line Curve formula without a graph in Microsoft Excel

I am trying to obtain a solution for the number where two curves meet. Specifically trying to determine the expected flow rate given a pump curve and system curve. The curves shift depending on variables that are read from a spreadsheet.

I can get the curve formulas by graphing them and requesting the trend-line feature, but then I have to manually enter the curve formulas, set them equal to each other, and solve the equation. Can Excel do this automatically from two data sets? Any help would be appreciated.

What kind curves are they (i.e., what kind of functions are those trend-line equations) - linear, polynomial, logarithmic, ....? Can you post sample trend line equations for the pump curve and the system curve.

It doesn't matter what kind of curves they are if they're both functions of a single, common variable (otherwise they couldn't be graphed together easily). All that's needed is one cell containing the common X value, one cell for each curve containing the f(x) and g(x) function values given the common X value and their other respective parameters, and finally another formula cell calculating the difference between the two function values.

Then use Solver to set that last cell to zero by changing the X value cell. If the two curve intersect transversely, Solver shouldn't have any difficulty finding the intersection point's X value, and both functions will return the same Y
value. If the two curve intersect non-transversely, e.g., two circles intersecting at a single point, then Solver might have some difficulty and the OP would need to play with its options.

Of course "Solver" wouldn't have any difficulty in finding the x- and y- values of the intersection point of f(x) and g(x) (by making the difference between the functions equal zero). However, please note that the solver solution is a one-time result, and doesn't automatically update when f(x) and g(x) change; so your have to manually invoke Solver every time the functions change (I know it is not a big deal, but is certainly an inconvenience). The OP did mention, "The curves shift depending on variables that are read from a spreadsheet...........

I can get the curve formulas by graphing them and requesting the trend-line feature, but then I have to manually enter the curve formulas, set them equal to each other, and solve the equation. Can Excel do this automatically from two data sets. My purpose of asking the OP for the functional forms of f(x) and g(x) was that, if both are linear, logarithmic, simple exponential, quadratic, or power functions (and not higher order polynomials), then the x-value (and from it, the y-value) of the intersection point can be obtained using analytical formulas. Since those formulas will reference the cells containing the parameters/constants of f(x) and g(x), the solutions will automatically (and conveniently) update when f(x) and g(x) change. If f(x) and g(x) are higher-order polynomials or if they are of different functional forms, analytical solutions may be difficult/impossible to obtain, and "Solver" is the only recourse.

If the OP's functions are simple low order polynomials, logarithmic or exponential functions to begin with, it'd be inefficient for the OP to be fitting trend lines. And the trend lines aren't necessarily good approximations for the underlying functions near the point of intersection.

Solver is a manual solution (unless the OP were to use a Calculate event handler to run Solver whenever the function parameters change upon any recalc), but it's likely to be more accurate than finding the intersection point of trend line approximations to the underlying functions.

The question, as I understood from the OP's posting, is not how to obtain f(x) and g(x) for two sets of real-life data; the OP says "I can get the curve formulas by graphing them and requesting the trendline feature,...". (Incidentally, the functions have got to be one of the types that the Trendline tool can handle, which are linear, exponential, logarithmic, polynomial, ...). The OP's question, on the other hand, was how to find the intersection point of f(x) and g(x). As I did mention in my earlier response, I do agree that Solver can certainly find the intersection point. However, my point is, if the solution for x when f(x)=g(x) can be expressed as an explicit analytical function (which is possible if both functions are linear, quadratic, simple exponential, logarithmic, or power functions, and not higher order polynomials), calculating the intersection point using such an analytical solution is certainly superior to finding an 'optimized' solution using Solver. Any optimization method, including Solver, is limited by the optimization criteria (algorithm used, convergence criterion, decimal tolerance, .....), and also on the initial values of the optimization parameters (in this case x-value of the intersection point). Of course, for simple functions such as the ones considered here, the Solver-result may be as good as the analytical solution, but the analytical solution is absolute. An added advantage with analytical solutions, as I had mentioned, is that the solutions update when the functions change (As you correctly pointed out, the same can be accomplished by using Calculate Event Handler to run Solver). A couple of questions about your remarks. My apologies if I haven't understood those remarks correctly. "If the OP's functions are simple low order polynomials, logarithmic or exponential functions to begin with, it'd be inefficient for the OP to be fitting trend lines." But, aren't those the only function-types that Trendline handles (besides moving-averages)? What type of functions would lend themselves to be efficient to be fit with trendlines? And, when the functions do belong to those simple types, how would you efficiently fit them to real data without using Trendline (or Solver for that matter). "And the trend lines aren't necessarily good approximations for the underlying functions near the point of intersection.

Likely to be more accurate than finding the intersection point of trend line approximations to the underlying functions." f(x) and g(x) are two functions that happen to intersect at some x,y-point when somebody plots them out on a common graph; crudely put, the functions don't even know the existence of each other (If the two functions are coupled/correlated it is a totally different story). Any uncertainty in finding the intersection point of f(x) and g(x) depends on the quality of how well each function fits its respective x,y-data and how those uncertainties add up near the intersection point. Calculating the intersection point using the analytical solution can not diminish the inherent uncertainty in the intersection point, but it would certainly not worsen it; on the other hand, an optimization method such as Solver could, in certain situations, add some extra uncertainty to the solution due to inherent limitations of the method. But then, when explicit analytical solutions are not possible for a system of equations, one has to resort to optimization methods.

Polynomial equations are the best fit. As I mentioned, the system pressure and pump output pressure varies by fluid properties. An example of a data set from one fluid is as follows:
GPM Pump Head System Head
0 2279 123
300 2254 145
800 2219 277
1300 2192 531
1800 2171 905
2300 2155 1399
2800 2141 2014
3300 2130 2750
3800 2118 3607
4300 2105 4584
4800 2089 5681
5300 2069 6900
5800 2042 8239
Graphing the data and selecting a 3rd order polynomial for the trendline gives the following equations:
System rate f(x): y=-5E-21x^3=0.002x^2-1E-13x+123.08
Pump rate g(x): y=-2E-09x^3=2E-05x^2-0.0896x+2279.4
The solution for flow rate is where the pump and system pressures are equivalent, or where the two equations are equal.

If I am not mistaken, I believe your equation results are slightly off because of display error. For example, your x^3 term in System Rate should be about 8.34 *10^-12 (vs your -5*10^-21). When Solving equations, you may find it better to extract the coefficients. For Example, select 4 horizontal cells and Array enter the following with X & Y referring to your data. =LINEST(Y ,X^{1,2,3}) (Ctrl+Shift+Enter). These coefficients appear correct when compared to the output from another program. (System): 8.34352E-12 ,0.000241208 ,1.19407E-05, 123.0624551 (Pump): -2.02024E-09 ,2.01471E-05, -0.089731972, 2279.045625. Subtract the coef. to get something like : -2.0285*10^-9*x^3-0.000221*x^2-0.08974*x+2155.98317. Solve the cubic equation (equation =0) to get three real solutions : -108474.58402 / -3389.100225 2890.94350 / Only 2890 appears as the feasible solution.

There are a few ways by which you can accomplish what you want. Let's suppose that column flow rates (GPM) are in A2:A14, Pump Pressure are in B2:B14, and System Pressure are in C2:C14. Create a column D214 as follows: In D2 enter the formula =B2-C2, and drag the formula down to D14.

Method 1 (Maybe Good!): By making an inverse plot (a slightly approximate, but very quick, solution). Make a plot of A2:A14 (Y-axis) and D214 (X-axis) and fit a polynomial (maybe, 3-rd order) and get the trendline equation. The constant term is what you want. Unfortunately, this result may be about 2-4% off from the 'correct' value. For example, for your data, the result from this approach is 2952 gpm (the trendline equation is y = 1E-08x3 - 0.0001x2 + 0.8402x + 2952.2) is about 2% off from the correct value (2891 gpm) A convenient aspect of this method is that the trendline equation would automatically update when your original data change.

Method 2 (Better?): You can use Solver, and solve for the flow rate for which the difference between Pump Pressure and System Pressure will be zero. Obtain the trendline equation for The result obtained would be very good (e.g., 2891 for your sample data). However the inconvenient aspect is that the result WILL NOT update when your original data change(s). You have to manually invoke Solver every time that happens.

Method 3 (Best!): Exploiting Circular Reference. Select a 1 Row x 4 Column area, and entering the following formula, =LINEST(D214,A2:A14^{1,2,3}), confirmed with CTRL-SHIFT-ENTER (for your sample data, 2.029E-09, 2.211E-04, 8.974E-02, -2.156E+03) In E2, enter an arbitrary flow rate (gpm) value, say 300. In F2, =1.00001*E2 IN G2, =a*E2^3+b*E2^2+c*E2+d (where a, b, c, and d are the actual values of the coefficients from the trendline equation or linest function) In H2, =a*F2^3+b*F2^2+c*F2+d
In I2, =(H2-G2)/(F2-E2). In J2, =(I2*E2-G2)/I2.

Now comes the interesting part. Go back to E2 and enter =J2. Excel will complain and say there is a Circular Reference. Ignore that. Go to "Tools" --> "Options" --> "Calculation" Tab, check "Iteration" --> "OK" Now Excel will accept the circular reference. You would notice that E2 and J2 have become equal. That is the value you want!

Your sample data yielded the following results: 2890.9 2891.0 0.0 0.0 1.4 2890.9The values will update when your original data change.