SAT Math you NEED to know: Algebra
Summary
TLDRThe 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
- đ Algebra is a key concept on the SAT, especially in the math portion.
- đ The slope of a line is essential for solving problems involving perpendicular lines.
- 𧟠When solving systems of equations, eliminate one variable to find the other.
- đ Use rise over run to calculate slope in linear equations and word problems.
- đ Perpendicular lines have opposite reciprocal slopes, a key concept for SAT problems.
- đ Word problems often break down into linear functions, so identify key variables (X, Y) early.
- đ Check your answers by plugging values back into the original equations.
- đ€ When solving word problems, break them down into manageable steps and identify key terms.
- âł Don't get tricked by extra information in word problems; focus on what's actually being asked.
- đ 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 lineâs 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
đ 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.
đą 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.
đĄ 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.
đ©âđ« Understanding On-Site vs. Online Training Courses
A word problem about an apprenticeâs 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.
đ 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 bumblebeeâs head width. The key takeaway is correctly interpreting and applying the formula.
đ 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.
Mindmap
Keywords
đĄAlgebra
đĄSlope
đĄPerpendicular Lines
đĄSystem of Equations
đĄReciprocal
đĄLinear Function
đĄWord Problem
đĄCoefficient
đĄRise over Run
đĄSubstitution
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
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