How to solve simultaneous equations?

How to solve simultaionsequations such as this,

f1(x1,x2) = x1^2 + x1*x2 - 10;

f2(x1,x2) = x2 + 3*x1*x2^2 - 57;

with and without using symbolic functions for f1 and f2. Please do explain the process.

[Image Analyst]

Do you mean you want to do it numerically? Like with some specific, discrete values for x? What range of x do you want to cover?

clc;    % Clear the command window.
fprintf('Beginning to run %s.m ...\n', mfilename);
close all;  % Close all figures (except those of imtool.)
clear;  % Erase all existing variables. Or clearvars if you want.
workspace;  % Make sure the workspace panel is showing.
format long g;
format compact;
fontSize = 20;
numPoints = 50;
X1 = linspace(-5, 5, numPoints);
X2 = linspace(-2.5, 2.5, numPoints/2);
[x1, x2] = meshgrid(X1, X2);
f1 = x1.^2 + x1.*x2 - 10;
f2 = x2 + 3*x1.*x2.^2 - 57;
subplot(2, 2, 1);
imshow(f1, [], 'XData', [min(x1), max(x1)], 'YData', [min(x2), max(x2)])
axis('on', 'xy')
impixelinfo
title('f1', 'FontSize', fontSize);
xlabel('x1', 'FontSize', fontSize);
ylabel('x2', 'FontSize', fontSize);
subplot(2, 2, 2);
imshow(f2, [], 'XData', [min(x1), max(x1)], 'YData', [min(x2), max(x2)])
axis('on', 'xy')
impixelinfo
title('f2', 'FontSize', fontSize);
xlabel('x1', 'FontSize', fontSize);
ylabel('x2', 'FontSize', fontSize);
diffImage = abs(f1 - f2);
subplot(2, 2, 3);
imshow(diffImage, [], 'XData', [min(x1), max(x1)], 'YData', [min(x2), max(x2)])
axis('on', 'xy')
impixelinfo
title('abs(f2 - f1)', 'FontSize', fontSize);
xlabel('x1', 'FontSize', fontSize);
ylabel('x2', 'FontSize', fontSize);
subplot(2, 2, 4);
surf(x1, x2, diffImage, 'EdgeColor','none')
title('f2 - f1', 'FontSize', fontSize);
xlabel('x1', 'FontSize', fontSize);
ylabel('x2', 'FontSize', fontSize);
g = gcf;
g.WindowState = 'maximized'
minValue = min(diffImage(:))
[rows, columns] = find(diffImage == minValue)
x1AtMin = X1(columns)
x2AtMin = X2(rows)
subplot(2, 2, 3);
hold on;
plot(x1AtMin, x2AtMin, 'r*', 'LineWidth', 2, 'MarkerSize', 30)
subplot(2, 2, 4);
hold on;
plot3(x1AtMin, x2AtMin, minValue, 'r*', 'LineWidth', 2, 'MarkerSize', 30)

[Medhanand Kalluri]

Like is there no tool to solve two equations? I've used fzero for a single variable equation (not system of equations) (but to use that I first entered the function as a string; then used str2func to convert it into a function; then used fzero).

1) Similar to fzero, is there an inbuilt function available? If there is how do we use it for simultaeous functions?

2) For symbolic functions (i.e. if we create f1 and f2 with x1 and x2 defined as sym variables) we can't use fzero, then how do we find the roots?

Answer 1

[Bjorn Gustavsson]

Using symbolic tools you have solve - look at the help and documentation for that function for examlpes. Your problem should reduce to a cubic polynomial so should be possible to check "semi-manually".