Literal Equations

Summary

In this Missmi Math Tutorials video, Miss Smith introduces literal equations, which involve solving for one variable amidst multiple variables in an equation. She emphasizes the importance of identifying the target variable and suggests treating extra variables as numbers to simplify the process. Through examples, she demonstrates how to isolate the desired variable by performing inverse operations and maintaining equation balance, resulting in a solution that may not be 'pretty' but is accurate. The video aims to demystify literal equations and equip viewers with the confidence to tackle them effectively.

Takeaways

• 📚 Literal equations involve solving for a specific variable amidst multiple variables in an equation.
• 🖍️ It's important to identify and highlight the variable you are solving for to keep your focus clear.
• 🔍 Pretend the variables are numbers to simplify the process of isolating the variable you're solving for.
• ⚖️ Balance the equation by performing inverse operations on both sides to cancel out terms.
• 🔄 When isolating the variable, move all other terms to the opposite side of the equation.
• 📉 Literal equations may not result in a 'pretty' answer, but the goal is to isolate the desired variable.
• 🔄 Division is used to eliminate coefficients in front of the variable you're solving for.
• 📝 Rewrite the equation in standard form to make it easier to identify and combine like terms.
• 📉 Constants and variables can be separated and simplified if possible.
• 📌 Treat special symbols like pi as a variable in the context of the equation.
• 🔗 Remember that what you do to one side of the equation must be done to the other to maintain equality.

Q & A

What is the main focus of the tutorial video by Miss Smith?

The main focus of the tutorial video is to teach how to solve literal equations, which involve more than one variable and require isolating the variable specified in the problem.

Why is it important to identify the variable you are solving for in a literal equation?

Identifying the variable you are solving for helps you focus on isolating that variable, making it easier to solve the equation by treating other variables as if they were numbers.

What strategy does Miss Smith suggest for dealing with multiple variables in a literal equation?

Miss Smith suggests pretending the variables are numbers and focusing on isolating the variable you are solving for, regardless of the presence of other variables.

Why does Miss Smith recommend getting the variable you are solving for alone on one side of the equation?

Getting the variable alone on one side simplifies the process of solving the equation, as it allows you to concentrate on the necessary operations to isolate that variable.

What does Miss Smith mean by treating variables as if they were numbers?

Treating variables as if they were numbers means performing mathematical operations on them without getting confused by their presence, just as you would with numerical values.

How does Miss Smith approach moving variables from one side of the equation to the other?

Miss Smith recommends performing the inverse operation on the variable to move it to the other side of the equation, ensuring that the operation is applied to both sides to maintain equality.

What is the significance of standard form in solving literal equations?

Standard form, where variables are placed before constants, helps in organizing the equation and makes it easier to identify like terms and perform necessary operations for solving.

Why is it necessary to perform the same operation on both sides of an equation?

Performing the same operation on both sides of an equation ensures that the equality is maintained, which is a fundamental principle in solving equations.

How does Miss Smith handle terms that cannot be combined in an equation?

If terms cannot be combined because they are not like terms, Miss Smith suggests rewriting the equation and proceeding with the operations that can be performed to isolate the variable of interest.

What does Miss Smith advise when dealing with fractions in literal equations?

Miss Smith advises multiplying both sides of the equation by the denominator to eliminate the fraction, which simplifies the process of isolating the variable.

How does the presence of the variable 'pi' affect the solving process in literal equations?

In the context of the video, 'pi' is treated like any other variable, and the focus remains on isolating the specified variable, with 'pi' being part of the operations needed for solving.

Outlines

00:00

📚 Introduction to Literal Equations

In this segment, Miss Smith introduces the concept of literal equations, which involve solving for a specific variable amidst multiple variables within a mathematical problem. The emphasis is on understanding how to isolate the variable of interest, even when faced with a more complex equation. The tutorial suggests a strategy of treating variables as if they were numbers to simplify the process. The example given involves moving terms across the equation to isolate 'x', demonstrating the importance of performing inverse operations to cancel out terms and achieve the desired variable on its own.

05:02

🔍 Isolating Variables in Literal Equations

This paragraph delves deeper into the process of isolating variables in literal equations. It discusses the strategy of moving terms to one side of the equation to focus on the variable that needs to be solved for, in this case, 'y'. The video script explains the importance of performing inverse operations, such as subtraction and division, to cancel out terms and simplify the equation. It also highlights the concept of treating all terms as if they were numbers, even when they include variables like 'pi', which is treated similarly to a variable in this context. The summary includes an example where the equation is manipulated to isolate 'y', and the process of simplifying the equation is described in detail, including the handling of fractions and the use of standard form.

Keywords

💡Literal Equations

Literal equations are mathematical expressions that contain more than one variable and require the solver to isolate a specific variable as instructed. In the context of the video, the theme revolves around solving these equations by treating additional variables as if they were constants, focusing on isolating the variable of interest. An example from the script is when Miss Smith instructs viewers to solve for 'x' despite the presence of variable 'b', demonstrating the process of isolating 'x' by treating 'b' as a number.

💡Variable

A variable is a symbol, often a letter, that represents an unknown or changeable value in mathematical expressions. In the video, Miss Smith emphasizes the importance of identifying the variable to solve for, such as 'x' or 'y', and using algebraic manipulation to isolate it. The script illustrates this with the equation involving 'x' and 'b', where 'x' is the variable of interest.

💡Isolate

To isolate a variable in an equation means to manipulate the equation so that the variable is alone on one side. This is a fundamental step in solving for a variable, as demonstrated in the video where Miss Smith shows the process of isolating 'x' by moving all other terms to the opposite side of the equation.

💡Inverse Operation

Inverse operations are mathematical actions that undo the effect of another operation. In the script, Miss Smith uses inverse operations, such as subtracting 'b' to cancel out a '+b', to simplify the equation and move towards isolating the variable. This concept is crucial for solving literal equations by ensuring terms cancel out.

💡Combining Like Terms

Combining like terms is the process of adding or subtracting terms in an equation that have the same variable raised to the same power. In the video, Miss Smith mentions the inability to combine '6' and 'M' or '-B' and 'M' because they are not like terms, highlighting the importance of recognizing like terms in the equation-solving process.

💡Divide

Division is a fundamental arithmetic operation used in algebra to simplify equations. In the context of the video, Miss Smith uses division to eliminate coefficients in front of variables, such as dividing by 'M' to isolate 'X', demonstrating a common technique for solving literal equations.

💡Standard Form

Standard form in algebra typically refers to writing equations in a way that variables are grouped together and constants are grouped separately, often with variables first and constants second. Miss Smith emphasizes rewriting equations in standard form to make the process of solving for a variable clearer and more systematic.

💡Pretend

In the video, Miss Smith suggests 'pretending' that variables are numbers as a strategy to simplify the process of solving literal equations. This approach helps to demystify the problem by treating additional variables as constants, making the equation appear less complex and more manageable.

💡Simplify

Simplification in algebra involves reducing an equation to its most straightforward form, often by combining like terms or reducing fractions. In the script, Miss Smith simplifies the equation by dividing terms by -4 to isolate 'y', demonstrating the importance of simplification in reaching the solution.

💡Pi (π)

Pi, represented by the symbol 'π', is a mathematical constant that represents the ratio of a circle's circumference to its diameter. In the video, Miss Smith encounters 'π' in an equation and treats it as a variable, showing that even constants like 'π' can be part of literal equations and need to be handled appropriately during the solving process.

💡Algebraic Manipulation

Algebraic manipulation refers to the various mathematical operations performed on equations to transform them into a simpler or more useful form. The video script provides several examples of algebraic manipulation, such as subtracting '3x' and dividing by '-4', to illustrate the process of solving literal equations.

Highlights

Introduction to literal equations and their purpose in testing the understanding of solving for a particular variable.

The concept of literal equations involving more than one variable and the strategy to solve for the specified variable.

Highlighting the importance of identifying the variable to solve for and using visual aids like color to focus on it.

The strategy of treating variables as numbers to simplify the process of isolating the desired variable.

The preference for placing the variable to be solved on the left side in literal equations for ease of solving.

Demonstration of moving variables and constants across the equation to isolate the desired variable.

Explanation of the inverse operation to cancel out terms and isolate the variable.

The process of dividing by a coefficient to isolate the variable and the importance of maintaining the equation's balance.

The challenge of not being able to combine unlike terms and the approach to rewrite the equation accordingly.

The method of simplifying the equation to get the variable alone, even if the final answer isn't a 'pretty' one.

A step-by-step walkthrough of solving a literal equation for variable 'y', including moving terms across the equation.

The importance of writing in standard form when rewriting equations to maintain clarity.

The process of dividing each term by a coefficient to isolate 'y' and simplify the equation.

The handling of negative numbers and the conversion of the equation to a positive form for simplification.

An example of solving a literal equation involving the constant pi, treating it like a variable.

The technique of multiplying by the denominator to eliminate fractions in an equation.

The final steps of isolating the variable 's' in an equation involving pi and demonstrating the cancellation of terms.

The conclusion emphasizing the basic rules of solving equations and the importance of not being tricked by extra variables.

Transcripts

[Music]

welcome to missmi math tutorials I'm

Miss Smith in this video we're going to

be talking about literal equations it

tests you on how well you know you're

solving to get to a particular variable

that you're solving for so literal

equations changes it up where you've got

more than one variable within a problem

now it'll tell you what it wants to

solve for so in number one you'll it

wants you to solve for x but you've got

some other variables thrown in there so

you're not going to be able to get a

really nice pretty you know xal 3 you're

not going to get an answer like that

it's going to be a little Messier but

your goal is really just to isolate the

variable that they're asking for so for

number one you'll see that they're

asking for X so I think it's a good idea

to go ahead and just either highlight or

Circle or use a color pencil or

something and just know what it is

they're wanting you to solve for so you

can really look at it and think okay how

can I get that X alone and really the

best strategy is just to pretend these

are numbers pretend they're not

variables I know it looks scarier when

there are letters extra letters thrown

in there but just pretend they're

numbers Pretend This B is really a six I

want to get the X alone the easiest

first step is going to be to get my B

over to this side now I told you in some

of the last videos I always like to get

the variable we're solving for alone on

the left when you're talking about

literal

equations it's really just your

preference so when I'm doing literal

equations I'm going to go with whatever

is easiest but definitely when we're

solving regular equations or

inequalities get the variable to the

left literal equations just kind of go

with whatever's easiest I think it's

going to be easiest to just leave the X

where it is in this case and get

everything else over to the left so

let's start with a B this is a plus b so

we need to subtract B right I want to do

the inverse operation the opposite

operation so that these cancel to zero

but what I do to one side I have to do

to the

other now you might say oh I I can't

combine that 6 minus B those aren't like

terms and you're right we can't so

that's okay we just rewrite them I just

write 6 minus B

equals and I still have my MX and

remember X is what I'm trying to solve

for so this is really M * X I know that

because they're hugging each other up

close so I need to

divide to split them up and I want the m

to be the one that goes away right I

want that to reduce to one what I do to

this side I have to do to this

side so you'll notice now now my X is

alone I can't combine 6 and M I can't

combine negative B and M none of those

are like terms so I'm just going to

rewrite this as x = 6 -

B over

M so remember I said we're not going to

get a final pretty answer we're just

really moving things around to get the

variable we want alone that's all this

is and my second one it wants to solve

this equation for y so I'm going to just

highlight my y so remember that's what

I'm trying to get

alone so in this case it's going to be

easier I think to leave the variable on

the left side and get everything else on

the right I'm just going to go with

whatever is easiest so instead of trying

to separate this4 and Y first let's move

this 3x over let's just keep it simple

so this is a plus 3x so I want to do the

inverse I want to subtract 3x I want

that to cancel to zero what I do to this

side of the equation I have to do to

this

side so let me bring down what I have

left I've got -4

y equals now I can't combine 12us 3x

those are not like terms I'm just going

to rewrite it and I'm G to be careful to

rewrite it in standard form so I'm going

to put the -3x first then the positive

12 remember in one of my first videos we

talked about standard form and having

your variables first and your constant

second and so looking back at here I

should probably go ahead and switch that

so I could rewrite this as NE b + 6 over

M so coming back over here I subtracted

my 3 x i I rewrote it because I couldn't

actually combine these two

terms now I want to still get my y alone

now that's a -4 * y so I want to do the

inverse I want to divide by -4 because I

want those to cancel to one now what I

do to one side I have to do to the other

side now some teachers prefer to write

this whole thing over4 you can totally

do that personally I like to show that

each

term is being divided by -4 I think it's

just kind of easier to see when we can

combine things and maybe simplify or

reduce them and we definitely can here

so let's reduce what we can I've got my

y alone which is what I

want could I reduce -3 over -4 um in

terms of the numbers no 34s is as low as

it goes but but remember that a negative

/ a negative is a positive so I can go

ahead and make this a positive 34

x now this is pos2 / -4 so that's going

to be

A3 so I was able to get my y

alone in this last example hopefully

you'll recognize this sign this is pi um

it's 3.1 4 so on so forth um so we just

represent that with the symbol pi don't

let the pi scare you we treat it in this

case kind of like it's just another

variable we want to in this case it's

wanting us to solve for S so I'm going

to highlight what I'm solving

for the

S and I want to get the S alone and this

in this case I'm going to keep the S

over on the right side I'm going to try

to get all this other stuff over to the

left one step at a time remember I told

you guys in a recent video that we can

easily get rid of a fraction by

multiplying by the denominator so in

this case my denominator is

360 If I multiply this whole thing by

360 those

cancel but what I do to this side I have

to do to this side so that means I also

have to multiply that a by 360

so let me bring down what I have left

I've got

360a

equals pi r 2 s so notice all I did was

I knocked out that denominator to make

this a little better to look

at so now I'm solving for S so what this

means is pi * r s Time s so if I want to

isolate the S I just have to

divide by the pi and the r s the pies

will cancel the r squares will cancel

now I'm just left with s but what I do

to this side I have to do to this side

I'm going to say that probably 20

million more times in this unit and for

the whole rest of this class so just get

used to it so let me rewrite what I've

got here and I want to make sure I

simplify if I can but looking at this I

know I can't simplify any of that none

of it's like terms so I'm going to bring

down my S

equal 360 a 360 * a over p r

2 so there would be my final answer and

I didn't Square my final answer there

okay so that is literal equations again

you're just using your basic rules of

solving it's just thrown a couple extra

variables in there to try to trick you

but don't let it this has been M Miss

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