Strategies to Solve Multi Step Linear Equations with Fractions

Summary

In this educational video, Anil Kumar introduces two strategies to solve equations involving fractions: cross multiplication and finding the lowest common multiple (LCM). He demonstrates these methods with eight examples, showing how to eliminate fractions and simplify equations to linear forms. The video is designed to clarify concepts and improve problem-solving skills, encouraging viewers to practice the techniques and check their solutions for accuracy.

Takeaways

• 📘 The video series by Anil Kumar focuses on solving equations with fractions, aiming to clarify concepts through eight examples.
• 🔢 Two main strategies are introduced: 'cross multiplication' and 'getting rid of fractions', which are essential for simplifying equations.
• ✖️ 'Cross multiplication' is a method where the denominator is multiplied by the term on the other side of the equation to eliminate fractions.
• 🔑 The 'getting rid of fractions' strategy involves finding the lowest common multiple (LCM) of the denominators to simplify the equation.
• 📝 It's recommended to pause the video, copy the questions, and then solve them one by one following the strategies discussed.
• 🔄 In cases with multiple terms, finding the LCM is crucial as it allows for the cancellation of fractions and simplifies the equation to a linear form.
• 🧮 An example given is solving \( \frac{x + 2}{3} = \frac{4}{1} \) by cross multiplication, resulting in \( x + 2 = 12 \) and thus \( x = 10 \).
• 🔄 The video demonstrates how to handle equations with different terms and denominators by multiplying through by the LCM to clear the fractions.
• 📉 For equations with variables in the denominator, the LCM is used to eliminate the variable and solve for the variable.
• 🔍 The video emphasizes the importance of checking solutions by substituting back into the original equation to ensure accuracy.
• 📈 The series is designed to build confidence in solving equations with fractions, with a call to action for viewers to apply the strategies and share their feedback.

Q & A

What are the two strategies mentioned in the video for solving equations with fractions?

The two strategies mentioned are cross multiplication and finding the lowest common multiple (LCM) to eliminate fractions.

How does cross multiplication help in solving equations with fractions?

Cross multiplication involves multiplying the denominator of one fraction by the numerator of the other fraction and vice versa, which helps to eliminate the fractions and simplify the equation.

What is the purpose of finding the lowest common multiple (LCM) in solving equations with fractions?

Finding the LCM allows you to multiply each term in the equation by the LCM, which results in a linear equation without fractions, making it easier to solve.

How do you check if the solution to a fraction equation is correct?

You can check the solution by substituting the value of the variable back into the original equation and verifying if both sides of the equation are equal.

In the video, what is the first example of an equation solved using cross multiplication?

The first example is \( \frac{x + 2}{3} = \frac{4}{1} \), which simplifies to \( x + 2 = 12 \) after cross multiplication, leading to the solution \( x = 10 \).

What is the significance of the distributive property when multiplying terms by the LCM?

The distributive property is significant because it allows you to multiply each term inside the parentheses by the LCM separately, which is necessary for eliminating the fractions in the equation.

How does the video demonstrate solving an equation with multiple terms and different denominators?

The video demonstrates solving such equations by first finding the LCM of the denominators and then multiplying each term by this LCM, which leads to a linear equation without fractions.

What is the importance of applying the same operation to both sides of an equation?

Applying the same operation to both sides of an equation is important to maintain equality, which is a fundamental principle in solving equations.

How does the video handle equations where the variable is in the denominator?

The video suggests finding the LCM that includes the variable in the denominator, multiplying each term by this LCM, and then solving the resulting equation.

What is the final advice given in the video for solving equations with fractions?

The final advice is to go through the examples again to reinforce understanding, and to apply the learned strategies to solve any equation involving fractions.

Outlines

00:00

📘 Introduction to Solving Equations with Fractions

Anil Kumar introduces a series on solving equations with fractions, emphasizing the importance of understanding two main strategies: cross multiplication and eliminating fractions by finding the lowest common multiple (LCM). He encourages viewers to pause the video to copy down eight example problems and then follows with a step-by-step guide on applying these strategies. The first strategy, cross multiplication, is demonstrated with an example where fractions are eliminated by multiplying both sides of the equation by the denominator. The second strategy involves finding the LCM of the denominators to clear fractions from the equation, transforming it into a simpler linear equation.

05:03

🔢 Applying Cross Multiplication and LCM Techniques

The video continues with Anil Kumar applying the cross multiplication technique to solve equations with fractions. He demonstrates how to multiply the entire equation by the denominator to eliminate fractions, resulting in a linear equation that's easier to solve. He then moves on to examples where the LCM strategy is necessary, such as when dealing with multiple terms with different denominators. Anil shows how to find the LCM of the denominators and multiply each term by this number to clear the fractions. The process is illustrated with detailed examples, including how to handle equations with three terms and how to correctly apply the LCM to each term to maintain the equation's integrity.

10:08

📐 Advanced Techniques for Fractional Equations

In this part of the video, Anil Kumar tackles more complex equations with fractions, focusing on finding the lowest common denominator (LCD) when the denominators are not straightforward multiples of each other. He explains the ladder division method for finding the LCD and demonstrates how to multiply each term of the equation by this number to eliminate fractions. The video illustrates the process with equations that involve distributing the LCD across terms and combining like terms to solve for the variable. Anil emphasizes the importance of correctly applying the distributive property and combining like terms to arrive at the solution.

15:09

🏁 Wrapping Up the Strategies for Fractional Equations

Anil Kumar concludes the video by summarizing the strategies for solving equations with fractions. He reiterates the importance of cross multiplication and finding the LCD or LCM to simplify equations. He encourages viewers to practice these techniques with the provided examples and to check their solutions for accuracy. The video ends with a call to action for viewers to engage with the content by leaving comments, sharing their views, and subscribing to the channel for more educational content. Anil expresses his gratitude for the viewers' time and wishes them well in their learning journey.

Keywords

💡Solving Equations

Solving equations refers to the process of finding the value(s) of the variable(s) that make the equation true. In the context of the video, this involves various strategies to simplify and resolve equations that contain fractions. The video aims to clarify these methods, making complex equations more approachable.

💡Fractions

Fractions are mathematical expressions that represent a part of a whole, expressed as a numerator divided by a denominator. The video focuses on solving equations with fractions, which can complicate the solving process due to the need to deal with different denominators and potentially complex arithmetic.

💡Cross Multiplication

Cross multiplication is a technique used to eliminate fractions in equations by multiplying both sides of an equation by the denominators of the fractions. In the video, this strategy is one of the methods introduced to simplify equations, making them easier to solve.

💡Strategy

A strategy in this context refers to a systematic approach or plan used to solve equations with fractions. The video outlines two main strategies: cross multiplication and finding the lowest common multiple (LCM). These strategies are crucial for understanding how to tackle different types of fraction-containing equations.

💡Lowest Common Multiple (LCM)

The LCM is the smallest number that is a multiple of two or more numbers. In the video, finding the LCM is presented as a strategy for dealing with equations that have multiple fractions with different denominators. Multiplying each term by the LCM helps to eliminate the fractions and simplify the equation.

💡Linear Equation

A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. The video demonstrates how to transform equations with fractions into linear equations by eliminating the fractions, making them simpler to solve.

💡Distributive Property

The distributive property is a fundamental algebraic principle that allows for the multiplication of a term by a sum or difference, distributing the multiplication over the addition or subtraction. In the video, this property is used when multiplying through brackets to clear fractions, as seen in the examples provided.

💡Denominator

The denominator is the bottom number in a fraction, indicating the total number of equal parts the whole is divided into. The video emphasizes the importance of dealing with denominators when solving equations with fractions, often aiming to eliminate them through the use of strategies like cross multiplication or finding the LCM.

💡Numerator

The numerator is the top number in a fraction, indicating the number of parts being considered out of the whole. While the video's primary focus is on denominators due to the nature of solving equations with fractions, the numerator also plays a role in the arithmetic operations performed during the solving process.

💡Check the Answer

Checking the answer is the process of verifying that the solution to an equation is correct by substituting the solution back into the original equation. The video suggests that this is a good practice to ensure that the steps taken to solve the equation have been accurate, although it may not be necessary in every case.

Highlights

Introduction to a series on solving equations with fractions.

Eight examples provided to illustrate the solving process.

Strategy one: Cross multiplication to eliminate fractions.

Strategy two: Finding the LCM (Lowest Common Multiple) to clear fractions.

Cross multiplication explained with an example equation.

Demonstration of solving an equation using cross multiplication.

Verification of the solution by substituting back into the original equation.

Application of both strategies to solve equations with multiple terms.

Explanation of how to find the LCM for denominators.

Step-by-step solution of an equation using the LCM strategy.

Use of the ladder division method to find the LCM.

Solving an equation with a variable in the denominator.

Technique for avoiding negative signs when solving equations.

Final example demonstrating the use of LCM with a variable in the denominator.

Encouragement for viewers to practice the strategies on their own.

Invitation for feedback and subscription to the video series.

Transcripts

I'm Anil Kumar welcome to my series on

solving equations and thanks a lot for

another request you want to understand

how to solve equations with fractions

here are eight examples and I hope by

the end of the video you will have all

your concepts absolutely clear I like

you to pause the video copy these

questions and then we are going to take

them one by one I'm going to use two

strategies here to solve all these

questions

let me call strategy one as cross

multiplication

and strategy to has get rid of fractions

now cross multiplication also helps to

get rid of fractions let's say the

number is three here we'll just take it

on the other side so we'll get rid of

three from the left side now sometimes

what happens is we have different

numbers right for example here we have

three terms here how do we get rid of

fractions so in this case we'll look for

the LCM lowest common multiple will

multiply both sides by lowest common

multiple so the strategy will be LCM

times both sides and then we will solve

as a linear equation without fractions

so I hope the strategy is absolutely

clear now let's apply the strategy to

all these questions and solve them one

by one now in such case which is kind of

a ratio you could do cross

multiplication

let's understand what we are trying to

do think like this we have our equation

as X plus 2 over 3 equals to 4 over

nothing so think we have one here so

cross multiplication means this 3 gets

multiplied with the term on the right

side and this term the whole term gets

multiplied by 1 you get an idea that

means it remains kind of same that's

what we mean

so cross multiplication we're left with

X plus 2 on the left side and on the

right side we get 4 times 3 or 3 times 4

1 of the same thing so what we have here

is X plus 2 equals to 12 and now this is

without fractions easy to solve x equals

2 12 minus 2 which is 10 so we get x

equals to 10 some of you can check the

answer by placing 10 what do we get we

get 10 plus 2 over 3 which is 12 over 3

which is indeed 4 so that works perfect

so you may or may not check for the time

being but it's a good practice so you

may check ok so I think the technique is

absolutely clear less applied once again

so we get 1 we get X minus 6 equals 2 2

times 3 so that is X minus 6 equals to 6

X is equals to 6 plus 6 so X is equals

to 12 and you can check your answer 12

minus 6 over 3 is 6 over 3 which is

indeed - so that's what you expected

correct okay

so we have another similar kind of a

question now here we'll apply both the

techniques so this one I think now you

can do easily we can write this as minus

3x equals to 4 times 5 so we have 2

minus 3x equals to 20 minus 3x equals to

20 minus 2 so minus 3x is equal to 18 X

is 18 divided by minus 3 wedges minus 6

so we get x equals 2 minus 6 perfect

next one now here what we see is that we

see three different terms so this

technique is not going to work we have

to find the lowest common multiple so in

this case what is the LCM so we are

using the second strategy we are finding

lowest common multiple for denominator

so this is for denominator so 3 6 & 2

the lowest common denominator is 6 so we

are going to multiply each term by 6

right so let's rewrite this so we get 4x

over 3 equals 2 the equation is 7 over 6

minus 5x over 2 we are going to multiply

each term by 6 on both the sides do you

see this now since you do the same

operation on both sides you don't change

the equation but you could simplify it

so so when you do so you could get rid

of fractions that's the whole idea right

so here 3 goes 2 times 6 one time and

that goes 3 time so basically you get

linear equation without fractions 4

times 2 is 8 so I grab this as a tax

equals to 7 minus 3 times 5 15 X correct

so 4 gets multiplied by 2 we get 8x

several remains our search + 5 - x 3 LCD

also we call lowest common denominator

so we will also call this lowest common

denominator and you get a linear

equation now it is simpler to solve

bring it to one side 8x + 15 x equals to

7 and when you add this 8 plus 5 is 13

so get 23 x equals to 7 so X is equal to

7 over 23 so that becomes the solution

for the given equation so I hope these

steps are absolutely clear so in this

page we have seen both the techniques

here we did cross multiplication and

then in D we found the lowest common

denominator which is actually the lowest

common multiple of the denominators

multiplying the same gets rid of the

denominator and then you can solve as

shown here so so I hope the strategy is

absolutely clear let's move on and take

a rest of the questions now I would like

you to pause this video apply the right

strategy and then solve so these two

questions we again have three terms not

just two terms right so what do you

expect in this case we have to find the

lowest common denominator right so let's

find the least common denominator so

lowest common denominator which is 6 - 4

is what as 12 right so is 12 now for

some of you how do we figure this out

let's take that now so to find this we

have a ladder division method you could

write all these numbers 6 2 & 4

/ some common factors let's say - in

this case we get 3 times 1 & 2 so 2 goes

3 times in 6 1 times in 2 2 times in

full so I left with these numbers so the

lowest common denominator I should say

multiple now because we found that is 2

times 3 times 1 times 2 which is 12 so

that is the technique of finding the

lowest common denominator

perfect so once we find that we have to

multiply by this number to all the terms

so we have X plus 7 over 6 plus so we

are going to multiply this by 12 so that

is this 12 now 12 times 1/2 equals to 12

times X minus 2 over 4 as expected all

the denominators can now be cancelled

with 12 so 6 goes 2 times 2 goes 6 times

4 goes 3 times now this 3 should be

multiplied to both remember this part

right otherwise you will get wrong

answer

same here 2 has to be multiplied with

both X and 7 so when you open this

bracket you get 2x plus 14 plus 6 equals

to 3x minus 6

I hope this step is clear we are

multiplying this 2 with both the numbers

right applying the distributive property

now you have to solve bringing the X

terms together and the constants

together sometimes we know 3x is higher

we will bring X to the right side we

have 14 plus 6 let me write 14 plus 6

this X coming this side will become it

positive 6 3x minus 2 X so we get x

equals 2 26 so x equals 2 26 is our

answer is that clear

now let's do the next one so here what

is the lowest common denominator 4 & 2

it is 4 right since 4 is multiple of 2

so we'll multiply everything by 4 so

what do we get we get 4 times 1 over 4x

plus 1 in brackets equals to 4 times x

over 2 plus 4 times 3 so here

4 & 4 cancels in this case we get 2

times here it is 4 times so when you

open the bracket you get X plus 1 equals

2 2 X plus 12 again I will take this X

to the right side now since I see that

2x is greater bring 12 on this side so

we have 1 minus 12 equals 2 2 X minus X

that gives me x equals 2 minus 11 so I

hope you appreciate this strategy of

avoiding the negative sign with X it

also helps saves time perfect now let's

move on and take the last two questions

I hope by now you have learned this

strategy apply and solve these two

questions then check with my solution

here 5 & 2 so what is the lowest common

denominator in this case the lowest

common denominator is 10 so we'll

multiply every term by 10 so what we get

here is 10 times 2 over 5 times 3x minus

1 equals to 4 times 10 minus 10 times

1/2 X plus 2

now when you simplify 10 goes to x with

5 and 5 times with 2 so 2 times 2 is 4

so we're at 4 times 3x minus 1 equals

240 minus

so when you open this bracket remember

to apply minus 5 into both right so

minus 5x and minus 10 so take care of

minus sign also right so you have to

multiply with both the terms bring XS

together let's open this bracket also so

we get 12 X minus 4 equals 2 let's

combine them 40 minus 10 is 30 minus 5x

bring 5x to the left side it 12x plus 5x

equals 230 plus 4 and that is 17 x

equals 234 or X is equals 234 over 17

which is too perfect so that is how

you're going to solve it now let's take

the last example this is kind of

different since you have X in the

denominator to see this part X is in the

denominator so in this case what is the

lowest common denominator it is 6 X you

see that not just 6 6 X so we are going

to multiply each term by 6 X so we have

6 X times 1 over 3 equals 2 6 x times 2

over X minus 6 x over 6 so that gives

you 6 divided by 3 is 2 so we get 2x

equals to 6 times 2 is 12 X and X cancel

rights we get 12 and here we get minus X

so see what happens X and X cancels this

goes to x + 6 and 6 cans so we get our

equation which is 2x equal to 12 minus X

now bringing X to the left we have 2x +

x equals to 12 3x equals to 12 X is

equal to 12 over 3 which is 4 so I hope

the solutions are absolutely clear that

is how it should be so

go through these examples once again so

we have different types of strategies

applied and these are sufficient to

answer or solve any equation involving

fractions feel free to write your

comments and share your views if you

really like and subscribe to my videos

that we create thanks for watching and

all the best