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 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

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.

Main content

Course: Algebra II (2018 edition) > Unit 10

Lesson 1: Solving equations by graphing

Solving equations graphically

Google Classroom

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

Introduction

Can you solve the equation

\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

\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.

$y = \log_{2} (x + 4)$ ‍

$y = 3 - x$ ‍

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

\log_{2} (x + 4) = 3 - x

x \approx 0.75

Reflection question

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

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

\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.

What is the solution of $2^{x} - 3 = (x - 6)^{2} - 4$ ‍?

x =

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rstarax
Posted 7 years ago. Direct link to rstarax's post “In the Introduction, it i...”
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?
Button navigates to signup pageComment on rstarax's post “In the Introduction, it i...”
(26 votes)
Answer
- ever.maggiotto
  Posted 4 years ago. Direct link to ever.maggiotto's post “I am wondering the same t...”
  I am wondering the same thing and would love it if someone were to explain this.
  Comment on ever.maggiotto's post “I am wondering the same t...”
  (9 votes)
MarcBaetiong0
Posted 7 years ago. Direct link to MarcBaetiong0's post “I literally went to desmo...”
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.
Button navigates to signup pageButton navigates to signup page
(6 votes)
Answer
- Karmanyaah Malhotra
  Posted 6 years ago. Direct link to Karmanyaah Malhotra's post “You should search Khan Ac...”
  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.
  Button navigates to signup page
  (4 votes)
Kyra Ng Jia Ni
Posted 8 years ago. Direct link to Kyra Ng Jia Ni's post “Is a graphic or a scienti...”
Is a graphic or a scientific calculator required for this?
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(4 votes)
Answer
JosieD
Posted 8 years ago. Direct link to JosieD's post “How do we use the graphs ...”
How do we use the graphs and why do we have to use graphs in math that I don't understand.
Button navigates to signup pageComment on JosieD's post “How do we use the graphs ...”
(2 votes)
Answer
- Garrett Shields
  Posted 8 years ago. Direct link to Garrett Shields's post “A graph of a function is ...”
  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.
  Comment on Garrett Shields's post “A graph of a function is ...”
  (6 votes)
Phạm Trần Minh Trí
Posted 4 years ago. Direct link to Phạm Trần Minh Trí's post “Is there a way to solve *...”
Is there a way to solve 2^x − 3 = (x - 6)^2 - 4 using algebra ?
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(3 votes)
Answer
- loumast17
  Posted 4 years ago. Direct link to loumast17's post “Not really, this would ev...”
  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
  Button navigates to signup page
  (3 votes)
J
Posted 4 years ago. Direct link to J's post “Hi, in the practice hint ...”
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.
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(3 votes)
Answer
Samuel Rodriguez
Posted 5 years ago. Direct link to Samuel Rodriguez's post “How can we 'quickly' iden...”
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?
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(2 votes)
Answer
Tjeerd Soms
Posted 7 years ago. Direct link to Tjeerd Soms's post “If the functions have to ...”
If the functions have to points where they interesect, which one is the correct solution?
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(1 vote)
Answer
- Shin Andrei
  Posted 7 years ago. Direct link to Shin Andrei's post “Both points can be a solu...”
  Both points can be a solution. This type of function is what we call many-to-one functions.
  Button navigates to signup page
  (1 vote)
00020296
Posted 5 years ago. Direct link to 00020296's post “How do I check to see how...”
How do I check to see how accurate I am on the calculator
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(1 vote)
Answer
Muhammad Farman Choudary
Posted 4 years ago. Direct link to Muhammad Farman Choudary's post “In the next quiz, how do ...”
In the next quiz, how do you find the other two points the lines passed through other than vertex?
For g(x):
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(1 vote)
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