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### Course: Algebra 2 (Eureka Math/EngageNY)>Unit 1

Lesson 13: Topic C: Lesson 32: Graphing systems of equations

# Solving equations graphically

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

## Introduction

Can you solve the equation ${\mathrm{log}}_{2}\left(x+4\right)=3-x$?
Would any of the algebraic techniques you've learned so far work for this equation?
Try as you may, you will find that solving ${\mathrm{log}}_{2}\left(x+4\right)=3-x$ 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.
$y={\mathrm{log}}_{2}\left(x+4\right)$
$y=3-x$
Now let's graph the equations.
Which of these best approximates the solution to the above system?

It follows then, that an approximate solution to ${\mathrm{log}}_{2}\left(x+4\right)=3-x$ is $x\approx 0.75$.

### Reflection question

Why does it follow that $0.75$ is a solution to the equation ${\mathrm{log}}_{2}\left(x+4\right)=3-x$?

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 ${\mathrm{log}}_{2}\left(x+4\right)=3-x$.
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 $y={2}^{x}-3$ and $y=\left(x-6{\right)}^{2}-4$ are shown below.
What is the solution of ${2}^{x}-3=\left(x-6{\right)}^{2}-4$?
$x=$

## Want to join the conversation?

• In the Introduction, it is mentioned that it is a "difficult task" to solve the equation algebraically. But difficult is not impossible. How could this be done?
• I am wondering the same thing and would love it if someone were to explain this.
• I literally went to desmos to graph the practice questions in this topic. Is that even allowed? If it's not, then how do I interpret where the line of a graph will go? It isn't stated in this topic, I might have missed a lesson.
• You should search Khan Academy for different types of equations. You will get graphing and all other aspects of them. I also recommend you to completely master your pre-algebra, algebra 1, and do the unit tests of all algebra 2 before this lesson. Then you will surely understand it.
• Is a graphic or a scientific calculator required for this?
• How do we use the graphs and why do we have to use graphs in math that I don't understand.
• A graph of a function is just a picture of all the x-y pairs that make the equation true, so if I want to know the x value when y=0, for example, I would graph the function using some kind of calculator, then look for all the points where the y-coordinate is zero (the x-intercepts), then whatever those x coordinates are, they are exactly the x values that will make y=0.
• Is there a way to solve 2^x − 3 = (x - 6)^2 - 4 using algebra ?
• Not really, this would eventually require logs, and even if it was as simple as 2^x = x it would turn into log_2(x) = x which isn't any easier to solve