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## Algebra (all content)

### Course: Algebra (all content) > Unit 5

Lesson 8: Solving equations by graphing (Algebra 2 level)- Interpreting equations graphically
- Interpreting equations graphically (example 2)
- Interpret equations graphically
- Solving equations graphically (1 of 2)
- Solving equations graphically (2 of 2)
- Solving equations graphically
- Solve equations graphically

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# Solving equations graphically

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

## Introduction

Can you solve the equation ${\mathrm{log}}_{2}(x+4)=3-x$ ?

Would any of the algebraic techniques you've learned so far work for this equation?

Try as you may, you will find that solving ${\mathrm{log}}_{2}(x+4)=3-x$ 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.

Now let's graph the equations.

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

### Reflection question

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 ${\mathrm{log}}_{2}(x+4)=3-x$ .

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 ${y=(x-6{)}^{2}-4}$ are shown below.

## 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?(25 votes)
- I am wondering the same thing and would love it if someone were to explain this.(8 votes)

- 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.(6 votes)
- 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.(4 votes)

- Is a graphic or a scientific calculator required for this?(4 votes)
- How do we use the graphs and why do we have to use graphs in math that I don't understand.(2 votes)
- 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.(6 votes)

- Is there a way to solve
**2^x − 3 = (x - 6)^2 - 4**using algebra ?(3 votes)- 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(3 votes)

- Hi, in the practice hint explanations when graphing g(x) I always find "A quick calculation shows that the graph of this function will also pass through (x,y) and (x,y)", however, I can't find anywere how you can make this calculations.(3 votes)
- How can we 'quickly' identify when an equation is going to be difficult to solve algebraically? and Why is log_2(x+4) = 3-x such a case? I mean which features could we eyeball that let us conclude that it's better to solve it graphically?(2 votes)
- If the functions have to points where they interesect, which one is the correct solution?(1 vote)
- Both points can be a solution. This type of function is what we call many-to-one functions.(1 vote)

- How do I check to see how accurate I am on the calculator(1 vote)
- In the next quiz, how do you find the other two points the lines passed through other than vertex?

For g(x):(1 vote)