Solving Rational Equation with Whole Number - Part 2 - General Mathematics

Summary

In this educational video, Ton teaches viewers how to solve rational equations with whole numbers. He starts by identifying the least common denominator (LCD) and restricted value (RV), which is crucial to avoid undefined expressions. The LCD in this case is x-5. He then demonstrates the process of eliminating the denominator by multiplying the entire equation by the LCD, leading to a simplified equation. After solving, he finds x=15, and checks the solution by substituting back into the original equation. The video concludes with a reminder to like, subscribe, and turn on notifications for more informative content.

Takeaways

• 📘 The video focuses on solving rational equations, specifically one involving a whole number.
• 🔍 The key to solving rational equations is eliminating the denominator.
• 🌟 The LCD (Least Common Denominator) for the given equation is (x - 5).
• ⚠️ The Restricted Value (RV) is x = 5, as this makes the denominator zero, resulting in an undefined expression.
• 📖 The process involves multiplying the entire equation by the LCD to eliminate the denominators.
• 🔢 After multiplying, the equation simplifies to 3x = 5x - 25.
• 🧮 Transposing terms leads to 2x = 30, which simplifies to x = 15.
• 📝 The solution set is x = 15, which is the value that satisfies the original equation.
• 🔄 To verify the solution, substitute x = 15 back into the original equation and check for equality on both sides.
• 📢 The presenter encourages viewers to like, subscribe, and turn on notifications for updates.

Q & A

What is the main topic of the video?

The main topic of the video is solving rational equations, specifically focusing on how to eliminate the denominator in a rational equation.

What is the given rational equation in the video?

The given rational equation is \(\frac{3x}{x-5} = \frac{5}{x-5}\).

What is the LCD (Least Common Denominator) for the equation presented?

The LCD for the given equation is \(x-5\), as it is the common denominator present in both fractions.

What is a Restricted Value (RV) in the context of the video?

A Restricted Value (RV) is a value that makes the denominator of a fraction equal to zero, which would make the fraction undefined. In this case, the RV is 5 because \(x-5 = 0\) when \(x = 5\).

How does the video suggest eliminating the denominator?

The video suggests eliminating the denominator by multiplying the entire equation by the LCD, which is \(x-5\) in this case.

What is the simplified form of the numerator after eliminating the denominator?

After eliminating the denominator, the numerator simplifies to \(3x\) on the left side and \(5x - 25\) on the right side.

How is the value of x found in the video?

The value of x is found by simplifying the equation to \(3x = 5x - 25\), then isolating x by subtracting \(3x\) from both sides to get \(2x = 30\), and finally dividing by 2 to find \(x = 15\).

What is the solution set for the rational equation according to the video?

The solution set for the rational equation is \(x = 15\).

How does the video suggest checking the solution?

The video suggests checking the solution by substituting \(x = 15\) back into the original equation to see if both sides are equal.

What is the result of checking the solution in the video?

Upon checking, the video confirms that substituting \(x = 15\) into the original equation results in both sides being equal, verifying that the solution is correct.

What is the final advice given by the presenter at the end of the video?

The presenter advises viewers who are new to the channel to like, subscribe, and enable notifications to stay updated with the latest uploads.

Outlines

00:00

📘 Introduction to Solving Rational Equations

The speaker, Ton, introduces the topic of solving rational equations with a specific example provided by a subscriber. The equation given is 3x/(x-5) = (5-5)/(x-5). Ton emphasizes the importance of eliminating the denominator to solve the equation. The LCD (Least Common Denominator) is identified as x-5, and the restricted value (RV), where the denominator becomes zero, is also x-5. The process involves multiplying the entire equation by the LCD to eliminate the denominator, resulting in a simplified equation of 3x = 5x - 25. The solution to the equation is found by isolating x, which gives x = 15 after dividing both sides by 2.

05:01

🔍 Verifying the Solution to the Rational Equation

Ton demonstrates how to verify the solution to the rational equation by substituting x = 15 back into the original equation. The left side of the equation simplifies to 45/10, and the right side also simplifies to 45/10, confirming that the solution x = 15 is correct. Ton concludes the video by encouraging viewers to like, subscribe, and turn on notifications for updates. He signs off with a friendly farewell.

Keywords

💡Rational Equation

A rational equation is an equation that contains one or more rational expressions, which are fractions where both the numerator and the denominator are polynomials. In the video, the main focus is on solving rational equations, specifically one that involves the expression 3x/(x-5). The equation is solved by eliminating the denominator, which is a key step in dealing with rational equations.

💡LCD (Least Common Denominator)

The Least Common Denominator is the smallest multiple that all the denominators of a set of fractions can divide into without leaving a remainder. In the context of the video, the LCD is identified as 'x - 5', which is used to eliminate the denominators in the rational equation. Multiplying each term of the equation by the LCD is a common technique to solve rational equations.

💡Restricted Value

A Restricted Value (RV) refers to the value that makes the denominator of a fraction equal to zero, as it would make the fraction undefined. In the video, the restricted value is calculated by setting the denominator 'x - 5' to zero, which results in x = 5. This value is crucial as it cannot be used as a solution to the equation.

💡Eliminate the Denominator

Eliminating the denominator is a process used in solving rational equations where the fractions are removed by multiplying both sides of the equation by the LCD. In the video script, this process is demonstrated by multiplying the entire equation by 'x - 5', which simplifies the equation and allows for solving for x.

💡Numerator

The numerator is the top part of a fraction, which represents the number of parts being considered. In the video, after eliminating the denominator, the numerators are simplified to form a new equation, '3x = 5x - 25', which is a step towards solving for x.

💡Transposing

Transposing in algebra involves moving a term from one side of an equation to the other, often involving changing its sign. In the video, transposing is used to move '3x' to the right side of the equation, resulting in '5x - 3x = 30', which simplifies to '2x = 30'.

💡Simplifying

Simplifying an equation involves reducing it to its most basic form by combining like terms and performing arithmetic operations. The video demonstrates simplifying the equation '5x - 3x = 30' to '2x = 30', which is a crucial step in solving for the variable x.

💡Solution Set

A solution set refers to the set of all possible values that satisfy an equation. In the video, after solving the equation, the solution set is found to be x = 15, which is the value that satisfies the original rational equation.

💡Checking

Checking is the process of substituting the found solution back into the original equation to verify its correctness. In the video, the solution x = 15 is checked by substituting it back into the original equation to ensure that both sides of the equation are equal.

💡Undefined

A fraction is undefined when its denominator is zero because division by zero is not defined in mathematics. In the video, it is emphasized that a fraction with a zero denominator is undefined, which is why the restricted value (x = 5) cannot be a solution to the equation.

Highlights

Introduction to solving rational equations with whole numbers.

Explanation of a rational equation given by the equation 3x/(x-5) = 5 - 5/(x-5).

Emphasis on eliminating the denominator in rational equations.

Identification of the LCD (Least Common Denominator) as x-5.

Definition and calculation of the Restricted Value (RV).

Explanation of why a fraction with a zero denominator is undefined.

Process of multiplying the entire equation by the LCD to eliminate the denominator.

Simplification of the equation after eliminating the denominator.

Isolation of the variable x by transposing terms.

Solving for x to find the value of x = 15.

Introduction to the solution set notation SS = 15.

Explanation of how to check the solution by substituting x back into the original equation.

Verification that the solution x = 15 satisfies the original equation.

Encouragement for viewers to like, subscribe, and use the bell button for updates.

Conclusion and sign-off by the presenter, Ton.

Transcripts

hi guys it's me ton in our today's video

we will talk about another part of

solving rational equation so we have

here 3x overx - 5 is equal to 5 - 5 /x -

5 so without further Ado let's do this

topic so let's solve this problem what

we have here is a rational equation

wherein actually this one is a requested

topic or video from one of our

subscribers video solving rational

equations with whole number so here it

is now guys the very important thing

here in solving rational equations is on

how to eliminate this

denominator first we need to know what

is your

LCD so as you can see here in this given

rational equation they'll C this simply

x - 5 because only have x - 5 we have

here invisible one as your

denominator

and we also have here another x - 5 so

in other words your LCD is simply x - 5

then after that we also need to

know your restricted value so so many of

us many of our viewers are asking what

is RV so it is

restricted

value so how do we calculate for the

restricted

value

guys it will become zero and here the

restricted value or

RV is simply from

because if you will plug in five here

and here your denominator will become

zero and

remember fraction with zero denominator

automatic that is

undefined okay so we are done with the

LCD and restricted value now let's use

the LCD to eliminate this denominator by

simply multiplying your whole equation

by x - 5

and x - 5 when I say whole equation and

close your equation parenthesis or

bracket and then multiply it both

sides here it will become this one we

have

3x

time x -

5

over x - 5 is equal to 5 5 then this

multiply here to your whole number it

will become 5

* x - 5 and here we have

minus multiply it here - 5 * x - 5 over

x - 5 so as you can see in

our numerator we have x - 5 x - 5 oh

sorry

okay so here as you can see we can

easily eliminate your

denominator and here your numerator so

it will become simply 3x okay then equal

to here which

is

5 5 *

X which is 5X then 5 * -5 that is -25 or

- 25 5 and here guys let's cancel

out 5 x - 5 and x - 5 we have - 5 so we

will simplify this equation it will

become

3x is = to

5x and this one

is3 so what will happen here is that I

would transpose 3x to the other side and

I will transpose -3 to the other side

so I will copy

5x the very reason

3x negative okay so 3x -3x then 30

postive 30 so 5x - 3x is simply 2x is

equal to 30 and to simplify and to

finally solve for the value of x what

you need to do is divide both sides of

the equation by two and and here it is

your X is equal to 15 and this is the

value of x here again this is the value

of x you can write down your answer as

solution set we have SS or solution set

be equal to this one

15 yeah now if you want to check whether

your answer is correct you can know you

can do a checking okay

checking so what will happen is that

using this original equation

we will substitute 15 to X so it will

become 3 * 15 okay 3 * 15 over 15 - 5 so

this is your X it will become

15 is equal to 5 okay 5 - 5 I'm copying

the original equation over this is 15 -

5

simplify 3 * 15 is 45

over this is 10 -

5 equal to 5

- 5 over

10 so as you can see this five can be

expressed as

50 / 10 - 5/

10 is = to 45 over 10 and what we have

here let me adjust

this one this is 45 / 10 and 50 10 - 5/

10 is simply 45 over 10 as you can see

the two sides of the equation upon

checking is are equal meaning the

variable X which is equal to 15 is

correct in the given equation 3x - 5 is

= to 5 - 5 over x - 5 so I hope guys you

learned something from this video on how

to solve for a rational equation with a

whole number so if you're new to my

channel don't forget to like And

subscribe but Bell button for you to be

updated latest uploads again it's me to

Chon byebye