Strategies to Solve Multi Step Linear Equations with Fractions
Summary
TLDRIn 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
📘 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.
🔢 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.
📐 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.
🏁 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.
Mindmap
Keywords
💡Solving Equations
💡Fractions
💡Cross Multiplication
💡Strategy
💡Lowest Common Multiple (LCM)
💡Linear Equation
💡Distributive Property
💡Denominator
💡Numerator
💡Check the Answer
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
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