~adapted from walch education Intersecting Graphs

In the graph of the functions f(x) and g(x), the set of solutions for f(x) = g(x) will be where the two graphs intersect. If f(x) = g(x) = b for a particular value of x, say a, then the point (a, b) will be on the curve defined by f and will also fall on the curve defined by g. Since (a, b) falls on both curves, they must intersect at point (a, b). Graphing Solutions of Functions

In the graphs of f(x) and g(x), look for the point(s) where the curves intersect. Substitute the x-value from the point into the functions to see if it is, or is close to, a solution. Note that it is possible for a system of equations to intersect at more than one point, at only one point, or to not intersect at all. Continued…

Making a table of values for a system of two equations means listing the inputs for each equation and then comparing the outputs. List the values of x to substitute into the first column. List the first equation in the second column and the corresponding outputs, which are the resulting values after substituting in the x-values chosen in the first column. Using a Table of Values to Find Solutions

List the second equation in the third column and the corresponding outputs, which are the resulting values after substituting in the x-values chosen in the first column. In the fourth column, list the difference of the first functions’ outputs minus the second equations’ outputs. Look for where the difference is the smallest in absolute value and for a sign change event. Continued…

A sign change event is where the values of f(x) – g(x) change from negative to positive (or positive to negative). The solution(s) to the system are where the sign change occurs and the difference in the two outputs is smallest in absolute value. If the difference between the outputs is 0, the value of x that was substituted is the x-coordinate of the solution, and the corresponding output is the y-coordinate of the solution. More…

Use a table of values to approximate the solutions for the following system of equations: f(x) = 3 x g(x) = 2 x + 1 Let’s see how it works.

Create the Table of Values There is a sign change from x = 0 to x = 2, and at x = 1, f(x) – g(x) = 0. This tells us the curves f and g intersect at x = 1.

~Dr. Dambreville Thanks for Watching!