Method

Solving an Equation Graphically

When finding the solution(s) to an equation, sometimes it can be difficult to solve the equation algebraically. In such cases, solving the equation graphically can be convenient.

\underline{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.

Two graphs intersecting a three different points

For a more concrete example, consider the following equation.

3 x = 2^{x} + 1

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

Create Two Functions From the Given Equation

Each side of the given equation forms a function.

3 x = 2^{x} + 1 \Rightarrow f (x) g (x) = 3 x = 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) = 3 x

is a vertical stretch of

y = x,

and

g (x) = 2^{x} + 1

is a vertical translation of

y = 2^{x} .

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.

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.

3 x = 2^{x} + 1

Substitute

$x = 1$

3 (1) = ? 2^{1} + 1

▼

Simplify right-hand side

ExponentOne

$a^{1} = a$

3 (1) = ? 2 + 1

MultByOne

$a \cdot 1 = a$

3 = ? 2 + 1

AddTerms

Add terms

3 = 3 ✓

Therefore,

x = 1

is an exact solution. The solution

x = 3

can be verified the same way.

3 x = 2^{x} + 1

Substitute

$x = 3$

3 (3) = ? 2^{3} + 1

▼

Simplify right-hand side

CalcPow

Calculate power

3 (3) = ? 8 + 1

Multiply

9 = ? 8 + 1

AddTerms

Add terms

9 = 9 ✓

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 : 3 x = 2^{x} + 1 x = 1 and x = 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.

	{{ 'ml-lesson-number-slides' \| message : article.intro.bblockCount }}
	{{ 'ml-lesson-number-exercises' \| message : article.intro.exerciseCount }}
	{{ 'ml-lesson-time-estimation' \| message }}

{{ article.displayTitle }}