SAT Math you NEED to know: Algebra

Summary

The video covers essential algebra concepts for the SAT, focusing on frequently asked and challenging questions. The presenter explains key topics like slope of perpendicular lines, solving systems of equations, and handling word problems involving linear functions, demand modeling, and equation interpretation. The video emphasizes breaking down word problems, identifying key information, and applying algebraic techniques like solving for variables and slope-intercept form. Viewers are encouraged to practice these concepts to improve their performance on the SAT's math section.

Takeaways

• 📘 Algebra is a key concept on the SAT, especially in the math portion.
• 📊 The slope of a line is essential for solving problems involving perpendicular lines.
• 🧮 When solving systems of equations, eliminate one variable to find the other.
• 📉 Use rise over run to calculate slope in linear equations and word problems.
• 📐 Perpendicular lines have opposite reciprocal slopes, a key concept for SAT problems.
• 📝 Word problems often break down into linear functions, so identify key variables (X, Y) early.
• 🔄 Check your answers by plugging values back into the original equations.
• 🤔 When solving word problems, break them down into manageable steps and identify key terms.
• ⏳ Don't get tricked by extra information in word problems; focus on what's actually being asked.
• 📏 Use the slope-intercept form (y = ax + b) for modeling relationships like price and demand in word problems.

Q & A

What is the importance of algebra in the SAT math portion?

Algebra is one of the key concepts on the SAT math portion. Many questions are frequently asked about algebra, making it crucial to understand and master it for success on the SAT.

How do you find the slope of a line that is perpendicular to another line?

To find the slope of a line that is perpendicular to another line, you take the opposite reciprocal of the original line’s slope. For example, if the slope of line K is -7/3, the slope of line J, which is perpendicular to K, would be 3/7.

What is a simple way to solve a system of equations like X + Y = 3.5 and X + 3Y = 9.5?

To solve this system, you can add the two equations together, allowing one variable to cancel out. In this case, the X terms cancel out, simplifying to find Y. Then, substitute the value of Y into one of the original equations to find X.

How can you check if your solution to a system of equations is correct?

You can check your solution by substituting the values of the variables back into the original equations. If the equations hold true with these values, your solution is correct.

What is the best approach to solving word problems involving linear functions?

The best approach is to break down the problem, identify the variables, and recognize that it follows a linear function pattern. Use the given information to find the slope and intercept, then solve for the unknown variable.

How do you calculate the slope when given two points, such as (40, 20000) and (60, 15000)?

The slope is calculated using the formula (Y2 - Y1) / (X2 - X1). For the points (40, 20000) and (60, 15000), the slope would be (15000 - 20000) / (60 - 40) = -5000 / 20 = -250.

In the equation 10X + 15Y = 85, what do the coefficients of X and Y represent?

In this equation, the coefficient of X represents the hours spent on on-site training courses (10 hours per course), and the coefficient of Y represents the hours spent on online training courses (15 hours per course).

How do you solve a word problem that asks for the difference in time between online and on-site training courses?

By comparing the coefficients of X and Y in the equation, you can see that an online course takes 15 hours and an on-site course takes 10 hours. Therefore, an online course takes 5 more hours than an on-site course.

How do you interpret the equation for the head width of a worker bumblebee based on body weight?

The head width (W) is estimated by adding 0.6 to 4 times the body weight (B) of the bee. The equation is W = 4B + 0.6. By substituting the body weight of 0.5 grams into the equation, the head width would be 2.6 millimeters.

What strategies can help with solving word problems on the SAT math portion?

The key strategies include breaking down the problem into smaller parts, identifying the variables, understanding the relationships between them, and solving step-by-step. Avoid getting overwhelmed by excess information and focus on what is being asked.

Outlines

00:00

📚 Introduction to SAT Algebra Essentials

The speaker introduces the topic of algebra on the SAT, emphasizing its significance in the math section. They highlight that the algebra problems discussed are taken from official SAT and College Board materials, focusing on the most frequently asked and difficult questions. The first example involves determining the slope of a line perpendicular to another, explaining the concept of opposite reciprocals.

05:02

🔢 Solving a Simple System of Equations

This section explains how to solve a system of linear equations, breaking it down into simple steps. The speaker focuses on isolating one variable by adding or subtracting the equations. They solve for both variables (x and y) and emphasize the importance of verifying the solution by substituting the values back into the original equations.

10:04

💡 Tackling a Word Problem: Linear Demand Function

The speaker tackles a word problem involving demand as a linear function of price. They walk through interpreting the variables, setting up a slope equation, and solving for the demand when the price is $55. This part demonstrates the importance of understanding the linear relationship between variables and systematically solving the problem using a clear equation.

15:06

👩‍🏫 Understanding On-Site vs. Online Training Courses

A word problem about an apprentice’s training hours is presented, where the speaker explains how to compare the hours required for on-site versus online courses. The focus is on understanding how coefficients in the equation represent the training hours, leading to the conclusion that online courses take 5 more hours than on-site courses. The speaker emphasizes how wordy problems can sometimes be simple in essence.

🐝 Bumblebee Head Width and Body Weight Problem

In this paragraph, the speaker explains another word problem, where the head width of a worker bumblebee is calculated based on its body weight. They demonstrate how to set up and solve the equation by substituting the given values into the formula, yielding the bumblebee’s head width. The key takeaway is correctly interpreting and applying the formula.

👍 Wrapping Up and Encouraging Engagement

The video concludes with the speaker thanking the audience and encouraging viewers to leave comments with any questions. They also invite viewers to subscribe and suggest specific math problems they would like to see in future videos.

Keywords

💡Algebra

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols. In the video, algebra is a key concept for the SAT math section, focusing on understanding how to solve equations and functions, including linear functions and systems of equations.

💡Slope

The slope of a line refers to its steepness and direction, calculated as the ratio of the rise (vertical change) over the run (horizontal change). In the video, the slope is crucial for solving problems involving linear equations, such as determining the perpendicular slope of a line.

💡Perpendicular Lines

Perpendicular lines are two lines that intersect at a 90-degree angle. The video explains how the slopes of perpendicular lines are opposite reciprocals of each other, as shown in the problem where the slope of line K is -7/3, and the perpendicular line J has a slope of 3/7.

💡System of Equations

A system of equations involves solving multiple equations that have the same variables. The video shows how to solve systems by elimination or substitution, such as the example with the equations x + y = 3.5 and x + 3y = 9.5, where the variables are solved step by step.

💡Reciprocal

A reciprocal is a number that, when multiplied by another number, results in 1. The video explains the concept of the opposite reciprocal in the context of perpendicular lines, where the reciprocal of -7/3 is 3/7 and their slopes are opposites.

💡Linear Function

A linear function is a mathematical function that creates a straight line when graphed, typically represented as y = mx + b. The video discusses how the SAT often includes questions involving linear functions, such as determining the relationship between variables like price and demand.

💡Word Problem

Word problems present math questions in a narrative format, requiring interpretation of text to form equations. The video emphasizes the importance of breaking down word problems, such as the example where demand and price are modeled as a linear function.

💡Coefficient

A coefficient is a numerical factor in a term of an algebraic expression. In the video, the coefficients of equations, like in 10x + 15y = 85, represent the number of hours for each type of training course, helping to solve how much longer an online course takes than an onsite course.

💡Rise over Run

Rise over run is a method for calculating the slope of a line, defined as the change in the y-values divided by the change in the x-values. The video uses this concept to find the slope in the word problem involving price and demand, where 5,000 over 20 gives a slope of 250.

💡Substitution

Substitution is a method used in algebra to replace one variable with an equivalent expression. In the video, substitution is shown when solving a system of equations, where the value of y is substituted into another equation to find the value of x.

Highlights

Introduction to the algebra concepts required for the SAT math portion.

Explanation of line K and line J, focusing on perpendicular slopes and the reciprocal relationship between them.

Detailed walkthrough of solving the slope of line K and identifying the perpendicular slope of line J.

System of equations example: Solving for x and y using simple algebraic manipulation.

Demonstrating how to add equations to cancel variables and find solutions for systems of equations.

Using substitution to check the solution in the system of equations for accuracy.

Explanation of a linear demand problem where price affects product demand, modeled as a linear equation.

Breaking down a word problem to identify key linear relationships between price and demand.

Finding the slope of a linear function using rise over run with specific price and demand points.

Solving for the demand based on price using linear function formulas in a real-world context.

Simplifying a word problem about training hours by interpreting the equation and solving for the difference in time between online and onsite courses.

Analyzing word problems with an equation format and recognizing key values in real-life scenarios.

Using model equations to estimate physical properties such as the head width of a bumblebee based on body weight.

Clarifying common mistakes in interpreting mathematical models within word problems.

Final message encouraging viewers to engage with the content and ask more math-related questions.

Transcripts

hi guys today we're going to be going

over the algebra you need to know for

the

SAT so I've pulled some of the most

frequently asked questions that are

going to be on the SAT and on the SAT

there's a large math portion and in this

math portion algebra is one of the key

Concepts you need to know in

master so I've grabbed these questions

from the official sat and college board

and they're ranked as the most

frequently asked and the most difficult

so these are definitely some of the key

Concepts you need to know let's get

started okay this first question is line

K is defined by y

= -7 3x + 5 line J is perpendicular to

line K in the XY plane what is the slope

of line a cut off but it would be K so

what is the slope of line K

so let's see let's analyze this first

what are the what is it really asking me

the slope and it also gives me the

information of line J is

perpendicular to line K now if you

remember a perpendicular line means that

their slopes would be the reciprocal of

one another and we already know the

slope of line K so this is already given

to us pretty much we know the slope is

-7 over

3x7 over 3 that's the slope of line K

and if line J is perpendicular that

means that it is the reciprocal the

opposite reciprocal of 17 over 3 which

would just be 3 over 17 it's also

important to note that it's not just the

reciprocal it's the opposite reciprocal

so you need to remember the negative

sign here and you need to switch it

because in a perpendicular line it's

also the opposite reciprocal so this one

is pretty straightforward but it's also

one of the key Concepts you need to know

in terms of linear functions and it's

just pretty much you need to remember

what perpendicular

means okay moving on to the next

one this one ISX + y = 3 3.5 and x + 3 y

= 9.5 solve the system again this is one

of another one of the super common asked

questions on the math portion this is a

simple this is a simple simple two this

is simple

this is a simple system of equations so

it's giving you two different equations

with two of the same variables but it's

multivariable and it's asking you to

find each of these

variables so what first pops into my

head is I need to isolate one of these

variables and this one actually is

pretty straightforward we can see

thatx and X

now this is great because they have the

same coefficient of one so that means

when we add them together these two will

cancel out cux + x = 0 so let's add

these equations together add

Sox + x = 0 y + 3 y = 4 Y and -3.5 + 9.5

= =

6 which means Y and now we can just find

y dividing by four on each side we get

six over 4 or 3 over two but for this

sake since they kept it in decimals I'm

going to turn this into a decimal too

1.5 okay so now we know y = 1.5 how do

we find X well we can just substitute

this back into this original equation to

get X so I'm going to use this first one

here because it's a little simpler so we

getx

plus 1.5 because we know that's the Y

now

equal

3.5 so this is just simple solve for x

we can subtract we need to isolate X so

minus 1.5 on both sides X = -5 and then

divide by1 on both sides x =

5 so now we can see that we have our two

variables and this would be our final

answer but if you have extra time on the

St you can always check your answers and

you should always check your answers so

that's what we're going to do to check

our answers we can just substitute this

back into the equation and see if it

equals what it's supposed to so let's

take the first one again

Sox so would be

-5 + Y which is 1.5

1.5 =

3.5 perfect cuz now these equal each

other and we know that our that we

solved for the variables

correctly and so overall this these

types of problems the main thing is

trying to cancel out one of these

variables and this one was pretty simple

but if I wanted to cancel out y first I

would need to multiply this equation by

three so that it' be 3 Y and we make

sure the coefficients are the same so

they cancel out but the main thing is

just canceling it out and then solving

for the first variable substituting and

finding the next

variable

okay next problem is a word problem so

an economist modeled the demand Q for a

certain product as a linear function of

the selling price P the demand was

20,000 units when the selling price was

$40 per unit and the demand was 15,000

units when the selling price was $60 per

unit based on the model what is the band

in units when the selling price is $55

per unit okay this one seems pretty

challenging at first especially since

it's a word problem and there's a lot of

numbers going around but you need to

break it down so let's look at what it's

saying so model the demand Q so demand

equals

q and it's a linear fun function this is

very key to this problem and like

selling price is p

so now that we know it's a linear

function and these are our two

variables um we can see that the way

that the price of P will be our X this

will actually be our X

variable because the price will be

dictating the demand and so that means

if this is causing this to change that

means this must be the Y

variable and we know that they'll it'll

be like this because it's a linear

function and linear functions are always

kind of built on this X and Y type

format so that means that they already

gave us the a bunch of numbers as we

continue to read on 20,000 units was the

selling price when the selling price was

$40 per unit so that means when it was

40 when X was was 40 the demand was

20,000 which is the Y and then we

continue on and the demand was $15,000

units when the selling price was $60 per

unit so that means that $60 must be the

X

variable and then 15,000 units must be

the

Y perfect so now it's saying based on

the model what is demanding unit when

the selling price is $55 per unit okay

so that means we know the selling price

is $55 per unit and we need to find when

what y will

be so now we have our equations okay

let's um I'm going to delete this so I

have a little extra

oh yeah let me delete this quickly to

have a little extra room to do this

problem

um so now that we have this we need to

find the equations that they gave us

from these two numbers so what first

pops into my brain even though it might

not be the fastest we can just do y = ax

+ B but we need to find the slope of

these two so the slope would just be

rise over run we can do

20,000

20,000 -

15,000 over 60 - 40 because that's the

slope formula if you don't remember and

so doing this quickly we can see that

this is 20,000 - 15,000 that's 5,000

over 20 so that would be

250 so that means our slope is

250 X plus b so now we need to solve for

b and to do this we can just substitute

again so we can do

um 4

20,000

equals 250 * 40 cuz that's the

X plus b so now doing this math quickly

we can see that this would equal 250 *

40 would

equal 10,000 and so 10 20,000 minus

10,000 so =

10,000 okay so now we know our equation

this would

be

10,000 okay now we can just solve so we

know X is

55 plus 10,000 and you solve for y now

this is pretty straightforward and I

don't want to do the math but you can

obviously as you when you solve this

you'll get the Y and you'll get one of

these answers

here now this one is definitely very

wordy and that that's one of the ways

they try to trick you but it's important

that when you saw me going through this

problem I was just go breaking down each

sentence and the stuff they were giving

me so the one of the mo most important

things here is you identify that says

linear here and that should give you a

clue oh it might be have to doing with

slope X and Y stuff like that and then

after you do that is you can they give

you most of the information and it's

just solving

then okay next up is another word

problem a certain Apprentice has

enrolled an 85 hours of training courses

the equation 10x + 15 y = 85 represents

the situation where X is the number of

on-site trading courses and Y is the

number of online training courses this

Apprentice has enrolled in how many more

hours does each online training course

take them on each on on-site training

course so this one is another very very

wordy one so let's see a certain

Apprentice has enrolled in 85 hours of

training courses the equation represents

the situation where X is the number of

on-site training courses and Y is the

number of online so X is onsite and Y is

online so how many more hours does each

online training course take than each

onsite training

course

well we can just see from these we need

to analyze it's trying to trick you

again because it's giving you all this

information but in reality you don't

really need all of it because it's just

asking you how many more hours does each

training course take than um does each

online one take then each

onsite so we know X is onsite and it's

asking how much longer online

takes

and we can see that these Coe fitions

actually represent how many hours each

of these courses take so I really don't

even have to solve for much because it

online takes 15 hours and onsite takes

10 hours which means that the online

training course takes five extra

hours and these are this is an example

of the SAT trying to trick you you

really cannot get trapped in all the

words it's giving you and you need to

focus on what is actually ask asking you

because sometimes it is more simple than

you think but they are just trying to

trick

you okay we have one more according to a

model the head width in millimeters of a

worker bumblebee can be estimated by

adding 6 to four times the body weight

of the be in Gs according to the model

what would the head width and

millimeters of worker B that has a body

weight of5 G another word problem and

this one

is um giving you the equations you need

so it says according to a model the head

width in millimeters of worker B can be

estimated so let's say head width is W

can be estimated by

adding 6 to 4 times so plus

0.6 to what to what 4 * the body weight

of the B so let's see it's four times

and we'll call the body weight

B okay so according to this model what

would the head width be of a body weight

of .5 G so now we can we already have

our equation and we can just substitute

and let's do that so 4 * .5

would just be 2 +

0.6 = W so this one's quite easy and W =

2.6

M this one is mostly about taking this

line here and being able to interpret it

and not getting confused because some

people might do like four plus plus

0.6 time B because it is confusing but

you need to break it down and know what

it's actually asking you

here okay thank you all for watching if

you have any more questions make sure to

comment them make sure to subscribe when

if you want to learn more about math and

comment on any specific math questions

you guys have I'll be continuing to do

videos like these thank you