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Method

# Solving an Equation Graphically

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

### 1

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. $3x = 2^x + 1 \quad \Leftrightarrow \quad 3x - 2^x = 1$

### 2

Graph the function

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

### 3

Identify any points with $y$-coordinate $C$

Now, find all the points on the graph that have the $y$-coordinate $C.$ For the example, the constant $C$ has the value $1,$ and there are $2$ points on the graph with the $y$-coordinate $1.$

### 4

Identify the $x$-coordinates

The $x$-coordinates of any identified points solve the original equation, $C = f(x).$ Note that solving an equation graphically does not necessarily lead to an exact answer. To verify a solution, substitute it into $f(x)$ and evaluate the expression. If the function value equals $C,$ it's an exact solution. If it's almost equal to $C,$ an approximate solution has been found. In the example, the $x$-coordinates are $1$ and $3.$

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

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

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