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Solving equations graphically

Learn a clever method for approximating the solution of any equation.


Can you solve the equation log2(x+4)=3x?
Would any of the algebraic techniques you've learned so far work for this equation?
Try as you may, you will find that solving log2(x+4)=3x algebraically is a difficult task!
This article explores a simple graphing method that can be used to approximate solutions to equations that cannot be solved directly.

Let's make a system

Thinking about the equation as a system of equations gives us insight into how we can solve the equation graphically.
So, let's turn the original equation into a system of equations. We can define a variable y and set it equal to the left and then the right side of the original equation. This will give us the following system of equations.
Now let's graph the equations.
Which of these best approximates the solution to the above system?
Choose 1 answer:

It follows then, that an approximate solution to log2(x+4)=3x is x0.75.

Reflection question

Why does it follow that 0.75 is a solution to the equation log2(x+4)=3x?
Choose 1 answer:

We can verify our solution by substituting x=0.75 into the given equation.

We did it!

Using the graphing method, we were able to solve the advanced equation log2(x+4)=3x.
We can use the graphing method to solve any equation; however, the method is particularly useful if the equation cannot be solved algebraically.

A general method for solving equations by graphing

Let's generalize what we did above.
Here is a general method for solving equations by graphing.
Step 1: Let y be equal to the expressions on both sides of the equal sign.
Step 2: Graph the two functions that were created.
Step 3: Approximate the point(s) at which the graphs of the functions intersect.
The x coordinate of the point(s) where the graphs of the functions intersect will be the solution(s) to the equation.

Try it yourself

Now let's put it all together. The graphs of y=2x3 and y=(x6)24 are shown below.
What is the solution of 2x3=(x6)24?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

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