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$Equation f(x)=g(x) $

To solve an equation, such as $f(x)=g(x),$ graphically, both sides should be graphed separately. Then, the $x-$coordinates of their points of intersection should be observed.
For a more concrete example, consider the following equation.
$3x=2_{x}+1 $

There are three steps to follow to solve the equation by graphing. 1

Create Two Functions From the Given Equation

Each side of the given equation forms a function.

$3x=2_{x}+1⇒f(x)g(x) =3x=2_{x}+1 $

The idea here is that the functions considered should be easy to graph. In this example, the functions created are some transformations of their parent functions. That is, $f(x)=3x$ is a vertical stretch of $y=x,$ and $g(x)=2_{x}+1$ is a vertical translation of $y=2_{x}.$ 2

Graph the Functions

Graph the functions in the same coordinate plane.

When the graphs of the functions do not intersect, the equation does not have a solution.

3

Identify the $x-$Coordinate of the Point(s) of Intersection

The solutions of an equation are the $x-$coordinates of any points of intersection of the graphs of the functions drawn. Since the graphs of the given equation intersect at two points, the equation has two solutions.

The $x-$coordinates of the points are $1$ and $3.$ Solving an equation graphically does not necessarily lead to an exact answer. To verify a solution, substitute it into the given equation and check if it produces a true statement. Therefore, $x=1$ is an exact solution. The solution $x=3$ can be verified the same way. Since the substitution resulted in a true statement, $x=3$ is also a solution of the equation. Therefore, the given equation has two solutions.$Equation:Solutions: 3x=2_{x}+1x=1andx=3 $

Note that if the point of intersection is not a lattice point, the exact solution may **not** be easy to find using this method.