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Solving an Equation Graphically


Solving an Equation Graphically

The function y=f(x)y=f(x) is comprised of all the (x,y)(x,y) points that satisfy its function rule. If the yy-coordinate is already given, the result is a one-variable equation such as C=f(x), C = f(x), where CC is the given yy-coordinate. It is then possible to use the graph of f(x)f(x) to solve the equation. For instance, the equation 3x=2x+1 3x = 2^x + 1 can be solved using this method.


If needed, rearrange the equation

If there are variables on both sides of the equation, the equation has to be rearranged so that the variables are on the same side. 3x=2x+13x2x=1 3x = 2^x + 1 \quad \Leftrightarrow \quad 3x - 2^x = 1


Graph the function

The side with the variables can now be seen as a function, f(x).f(x). Graph that function in a coordinate plane. For the equation 3x2x=1, 3x - 2^x = 1, the function is f(x)=3x2x.f(x) = 3x - 2^x.


Identify any points with yy-coordinate CC

Now, find all the points on the graph that have the yy-coordinate C.C. For the example, the constant CC has the value 1,1, and there are 22 points on the graph with the yy-coordinate 1.1.


Identify the xx-coordinates

The xx-coordinates of any identified points solve the original equation, C=f(x).C = f(x). Note that solving an equation graphically does not necessarily lead to an exact answer. To verify a solution, substitute it into f(x)f(x) and evaluate the expression. If the function value equals C,C, it's an exact solution. If it's almost equal to C,C, an approximate solution has been found. In the example, the xx-coordinates are 11 and 3.3.

Verifying the solution x=1x = 1 is done by evaluating f(1).f(1).

f(x)=3x2xf(x) = 3x - 2^x
f(1)=3121f({\color{#0000FF}{1}}) = 3 \cdot {\color{#0000FF}{1}} - 2^{\color{#0000FF}{1}}
f(1)=32f(1) = 3 - 2
f(1)=1f(1) = 1

We find that the function value is 1,1, which is the same as the value of CC for this equation. Thus, it is an exact solution. The solution x=3x = 3 can be verified the same way.