Linear Equations - Algebra
Summary
TLDRThis educational video script offers a comprehensive review of linear equations, focusing on three key forms: slope-intercept, standard, and point-slope. It explains how to calculate slopes using the rise-over-run method and identifies x and y-intercepts. The script also explores the concepts of parallel and perpendicular lines, their slopes, and provides step-by-step instructions for graphing linear equations in various forms, including slope-intercept, standard, and point-slope forms. It concludes with practical examples and exercises to solidify understanding.
Takeaways
- π The video is a review of linear equations, aimed at helping students prepare for tests.
- π There are three main forms of linear equations: slope-intercept form (y = mx + b), standard form (ax + by = c), and point-slope form (y - y1 = m(x - x1)).
- π The slope-intercept form is characterized by 'm' representing the slope and 'b' representing the y-intercept.
- π In the standard form, 'a', 'b', and 'c' are coefficients, and 'x' and 'y' are variables.
- π The point-slope form provides the slope (m) and a specific point (x1, y1) on the line.
- π’ The slope is calculated as the rise over the run, which can be positive, negative, or zero depending on the direction of the line.
- π A line with a 45-degree angle has a slope of one, and the slope increases as the line becomes steeper.
- βοΈ Horizontal lines have a slope of zero, and vertical lines have an undefined slope.
- π To find the slope between two points (x1, y1) and (x2, y2), use the formula (y2 - y1) / (x2 - x1).
- π X-intercepts are points where y = 0, and y-intercepts are points where x = 0.
- π Parallel lines have the same slope, and perpendicular lines have slopes that are negative reciprocals of each other.
Q & A
What are the three forms of a linear equation mentioned in the video?
-The three forms of a linear equation mentioned are slope-intercept form (y = mx + b), standard form (ax + by = c), and point-slope form (y - y1 = m(x - x1)).
What does 'm' represent in the slope-intercept form of a linear equation?
-In the slope-intercept form of a linear equation (y = mx + b), 'm' represents the slope of the line.
What is the slope of a line that rises 4 units and runs 3 units?
-The slope of a line that rises 4 units and runs 3 units is calculated as rise over run, which is 4/3.
What is the relationship between the slopes of two parallel lines?
-The slopes of two parallel lines are equal, meaning if one line has a slope of 'm', the other line will also have a slope of 'm'.
How do you find the slope of a line given two points (x1, y1) and (x2, y2)?
-You can find the slope of a line given two points by using the formula: slope (m) = (y2 - y1) / (x2 - x1).
What is an x-intercept and how do you find it?
-An x-intercept is a point where the line crosses the x-axis, which means the y-value is zero. To find the x-intercept, set y to zero in the equation and solve for x.
What is a y-intercept and how is it related to the 'b' value in the slope-intercept form?
-A y-intercept is the y-coordinate of the point where the line crosses the y-axis, which occurs when x is zero. In the slope-intercept form (y = mx + b), 'b' represents the y-intercept.
How do you graph a linear equation in slope-intercept form?
-To graph a linear equation in slope-intercept form, first plot the y-intercept, then use the slope to find another point by moving up by the rise and to the right by the run, and finally draw a line through the points.
What is the slope of a horizontal line?
-The slope of a horizontal line is zero because it does not rise or fall as it moves from left to right.
What is the slope of a vertical line?
-The slope of a vertical line is undefined because it represents an infinite rate of change as it moves from bottom to top without a horizontal displacement.
How do you determine if two lines are perpendicular?
-Two lines are perpendicular if the product of their slopes is -1, meaning if one line has a slope of 'm', the other line will have a slope of -1/m.
What is the process of graphing a linear equation in standard form?
-To graph a linear equation in standard form, find the x-intercept by setting y to zero and solving for x, find the y-intercept by setting x to zero and solving for y, plot these intercepts, and then draw a straight line through them.
How do you graph a linear equation in point-slope form?
-To graph a linear equation in point-slope form, identify the point (x1, y1) and the slope 'm'. Plot the point, then use the slope to find another point by moving up by the rise and to the right by the run, and draw a line through these points.
What does it mean if a linear equation is given in the form y = a constant?
-If a linear equation is given in the form y = a constant, it represents a horizontal line at the y-value of the constant, with a slope of 0.
What does it mean if a linear equation is given in the form x = a constant?
-If a linear equation is given in the form x = a constant, it represents a vertical line at the x-value of the constant, with an undefined slope.
Outlines
π Introduction to Linear Equations
This paragraph introduces the topic of linear equations, focusing on three key forms: slope-intercept form (y = mx + b), standard form (ax + by = c), and point-slope form (y - y1 = m(x - x1)). The slope (m) is explained as the rise over run, indicating the steepness of the line, while b represents the y-intercept where the line crosses the y-axis. The paragraph sets the stage for a deeper dive into the concepts needed for understanding and solving linear equations.
π Understanding Slope and Intercepts
The second paragraph delves into the specifics of calculating the slope using the rise over run method and understanding the implications of positive and negative slopes on the direction of the line. It also explains the concepts of x-intercepts (points where y = 0) and y-intercepts (points where x = 0), providing examples to illustrate these concepts. The paragraph concludes with a practical example problem to identify x and y-intercepts from a set of points.
π Graphing Linear Equations and Identifying Intercepts
This paragraph discusses the process of graphing linear equations, starting with identifying the slope and y-intercept from the equation. It provides a step-by-step method to plot points using the slope and then connect them to form the line. The paragraph also explains how to find x and y-intercepts from standard form equations and uses examples to demonstrate the graphing process, including identifying points where the line crosses the axes.
π Parallel and Perpendicular Lines
The fourth paragraph explores the relationship between parallel and perpendicular lines, explaining that parallel lines have equal slopes while perpendicular lines have slopes that are negative reciprocals of each other. It provides examples to illustrate how to determine if lines are parallel or perpendicular based on their slopes and introduces the symbols used to denote these relationships.
π€ Practice Problems on Slopes and Intercepts
This paragraph presents practice problems that require the application of concepts learned about slopes, x-intercepts, and y-intercepts. It includes examples of calculating the slope from two given points and determining the slopes of parallel and perpendicular lines. The paragraph serves as a practical application of the theoretical knowledge discussed earlier in the script.
π Graphing Equations in Various Forms
The sixth paragraph focuses on the techniques for graphing linear equations in different forms, including slope-intercept, standard, and point-slope forms. It explains how to identify key components from each form and use them to plot points and draw the line. The paragraph also addresses how to graph equations that represent horizontal and vertical lines, emphasizing the undefined slope for vertical lines and zero slope for horizontal lines.
π Conclusion and Practice with Multiple Choice Questions
The final paragraph wraps up the video script with a brief mention of practicing with multiple choice and free response questions to prepare for tests. It also includes an example of a multiple choice question that tests the viewer's ability to identify the correct graph for a given linear equation, emphasizing the importance of understanding slope and y-intercept.
Mindmap
Keywords
π‘Linear Equation
π‘Slope-Intercept Form
π‘Standard Form
π‘Point-Slope Form
π‘Slope
π‘X-Intercept
π‘Y-Intercept
π‘Parallel Lines
π‘Perpendicular Lines
π‘Graphing
Highlights
Introduction to linear equations and their forms, particularly for test preparation.
Explanation of the slope-intercept form of a linear equation: y = mx + b.
Description of the standard form of a linear equation: ax + by = c.
Introduction to the point-slope form of a linear equation: y - y1 = m(x - x1).
Discussion on the concept of slope and its calculation using the rise over run method.
Illustration of slope calculation with examples of rising and falling lines.
Explanation of the relationship between line direction and slope: positive for rising, negative for falling, zero for horizontal.
Clarification that the slope is undefined for vertical lines.
Method for calculating the slope using two points: m = (y2 - y1) / (x2 - x1).
Definition and identification of x-intercepts and y-intercepts in linear equations.
Example problem to identify x and y-intercepts from given points.
Difference between parallel and perpendicular lines and their slopes.
Explanation that parallel lines have the same slope, while perpendicular lines have slopes that are negative reciprocals.
Guidance on graphing equations in slope-intercept form using the slope and y-intercept.
Method for graphing equations in standard form by finding x and y-intercepts.
Technique for graphing equations in point-slope form using a given point and slope.
How to graph horizontal and vertical lines, and understanding their slopes.
Practice problems for graphing linear equations in various forms.
Transcripts
in this video we're going to do a review
of linear equations
it's especially for those of you who
have a test that you're studying for
so
let's begin by writing down some notes
there are three forms in which you can
write a linear equation
the first one
is known as the slope
intercept form
in the slope intercept form the linear
equation is written this way
y
is equal
to mx plus b
m
represents the slope which we'll talk
about later
and b
that represents the y-intercept
we'll also
talk more about that later as well but
for now
you want to write this equation so this
is the slope-intercept form
of a linear equation
now the next form that you want to be
familiar with
is
the standard form
so to write a linear equation in
standard form
this is how it's going to look like
it's a x
plus b y
is equal to c
a b and c are simply coefficients
x and y are the variables
but when written in that form it's
called
the standard form
the next form that you need to be
familiar with is
this one the point-slope form of a
linear equation
it's y minus y one
is equal to m
times x minus x one
as the name implies this equation can
tell you the slope and the point
the slope is the value of m
so whatever number you see here
and that's the slope
the point
is x1 comma y1
now let's talk about the slope
the slope is equal to the rise
divided by the run
so let's say if you have a linear
equation
that is rising
the slope is going to be positive
and let's say you have two points on
this line
now to go from the first point
to the second point
let's say it takes you have to travel
up
by four units
this is not drawn to scale by the way
this is just an illustration so let's
say you travel up four units
and then you travel three units to the
right
so in this case your rise is four
your run is three so rise over run the
slope would be four over three
and because
it's going up the line is going up the
slope is going to be positive
now here's another example
let's say
to go from this point to that point
we need to go down
three units
so the rise is negative because we're
going down so let's say negative three
units
let's say we have a run of positive five
it's positive because we're moving to
the right
the run should always be positive
now for this one the slope is going to
be rise over run
the rise is negative 3 the run is 5 so
it's going to be negative 3 over 5.
so because
the line is going down the slope is
negative
so that's a quick and simple way to
calculate the slope
using the rise over run method
now whenever a line
it goes up at a 45 degree angle
the slope of that line is going to be
one
if it goes up like this it's about one
half
and if it goes up even steeper let's say
like this
this would be a slope of two
so this line is very steep compared to
the other ones
now if the line is horizontal
the slope
is going to be zero
if it goes down at a 45 degree angle
like this
the slope is negative one
here it's about negative a half
and here
negative two
so any time the slope increases i mean
anytime the line increases the slope is
going to be positive
if the graph is going down the slope is
negative
and for any horizontal line the slope is
zero
so if we have a line that's going to the
right or to the left the slope is zero
for a vertical line
the slope is undefined
as the line becomes more vertical the
slope increases
eventually it can go up to infinity
and at some point it will be undefined
so just know that so if you have a
vertical line the slope is undefined for
a horizontal line the slope is zero
now you can calculate the slope of a
line
if you know the two points
so let's say if the first point
is x one comma y one
and the second point
is x2 y2
the slope of the line is going to be
y2
minus y1
divided by
x2
minus x1 so here's an example let's say
the first point
is two comma five
and the second point
is let's say five
fourteen
go ahead and calculate the slope
so y2 is going to be 14
and let's replace y1 with five
x2 is five
x one is two
fourteen minus five is nine
five minus two is three
nine divided by three is three
so the slope
of the line that connects
these two points
is equal to 3.
now let's talk about
x and y-intercepts
what is an x-intercept
and what is
a y-intercept
what do you think
the answer to that question is
an x-intercept is a point
but it's a specific point
the x-intercept is the point
where y is equal to zero
so let's say you have the point three
comma zero
at this point x is three
y is zero
this particular point is an x intercept
because
the y value is zero
so the x-intercept is any value of x
when y is zero
another example of an x-intercept is the
point
negative five zero
so the x-intercept in this case will be
negative five
if we have the point
two comma zero
the x intercept is two
so any point where the y value is zero
the x value is the x intercept
for the y-intercept the situation is
reverse
the y-intercept is the y-coordinate of a
point when x is 0.
so let's say if we have the point
0
4
the y-intercept is 4.
so remember when dealing with linear
equations the y-intercept is also equal
to b
so we would say that b
is 4.
here's another example of a y-intercept
let's say the point zero negative three
so x is zero y is negative three the y
intercept
is negative three so we can say that b
is equal to negative three
now let's summarize what we've just
learned
the x intercept
is the x coordinate
of a point
that contain a y value of zero
so as we see here
this point has a y value of zero the x
intercept is the x-coordinate of that
point
so x is negative five
the y-intercept
is the y-coordinate of a point that has
an x-value of zero
so for this point
the x value is 0 but the y intercept is
the y coordinate of that point so it's y
equals 4 or b equals 4.
so that's the basics of the x and y
intercepts
so here's an example problem for you
consider
these four points the point two comma
five
negative three comma zero
one comma two
and zero comma six
given these four points identify the x
and y-intercepts
so this point here has a y value of zero
therefore that point represents the
x-intercept
the x-intercept is specifically the
x-coordinate of that point so the
x-intercept
we could say it's x equals negative
three
but it you can also say that the
x-intercept is the point negative three
comma zero you can describe it both ways
the y-intercept is the point where x is
zero so this would be the y intercept
now you can say the y intercept is y
equals six
or you could say it's a b equals six
now the next thing that we need to
review
are parallel lines
and perpendicular lines
so what's the difference between
parallel lines and perpendicular lines
what would you say
and how do their slopes
relate to each other
parallel lines
they travel in the same direction
let's call this line one and line two
let's say the slope of line one has a
value of two
if the slope of line one has a value of
two and if line two is parallel to line
one
the slope of line two will be the same
it will also be two
what you need to know is that parallel
lines they have the same slope
so m1 is going to be equal
to m2
you can also describe the relationship
between two lines that are parallel
using the symbol
if you see this
double vertical line
it means that the two lines are parallel
so this is saying l1 is parallel to l2
but
for linear equations if you have a test
we need to know is that parallel lines
they have the same slope
now let's talk about perpendicular lines
let's say we have this line
which we'll call l1
and then this line l2
perpendicular lines
they intersect at right angles that is
at 90 degrees
the slope of the perpendicular line is
the negative reciprocal of the original
line
so let's say the slope of line one
let's say it's positive
three over four
the slope of line two is going to be the
negative reciprocal so you've got to
change the sign from positive to
negative
and you've got to flip the fraction
so it's going to be negative
4 over 3.
so m1 is going to be
negative
1 over m2
so that's the relationship between the
slopes
of two perpendicular lines
so we could say l1
is perpendicular this is the symbol for
perpendicular l1 is perpendicular to l2
you can see these two lines meet at
right angles
so that's how you can describe two
perpendicular lines
so remember the slopes
are negative reciprocals of each other
now let's work on some example problems
let's say that line one
is parallel
to line two
and let's say that you're given the
slope of line one
let's say that
the slope of line one is negative three
what is the slope of line two
if they're parallel the slope of line
one is equal to the slope of line two
therefore the slope of line two will be
negative three
now let's say that line one
is perpendicular to line two
and let's say you're given the slope of
line one let's say
it's
negative four
over seven
what is the slope of line two
the slope of line two
is going to be the negative reciprocal
of the slope of line one
so the first thing you need to do is
change the sign from
negative
to positive
and then you need to flip the fraction
from four over seven to seven over four
so that's going to be the slope
of the line that's perpendicular to the
first line
now let's talk about how we can graph
equations
in slope intercept form
so let's say we have the equation y
is equal to 2x minus 4.
how can we graph this equation
so first let's put in some
marks on a graph
feel free to try this example if you
want to
so the first thing we need to identify
is the slope and the y intercept so this
is in
y equals mx plus b form it's in slope
intercept form
so we can see that the slope
is equal to two
and we can see that the y-intercept
is negative four
with this information we have everything
that we need in order to graph this
function
so here's negative four
let's go ahead and plot the y-intercept
and then from the y-intercept we can get
the second point by using the slope
so the slope is 2
which means that it's 2 over 1.
so the rise is two the run is one so to
get the next point
we're going to go
up two units
and then travel one unit to the right
so that will give us the point
one negative two so we have an x value
of one
and a y value of negative two
now let's go up two and over one again
so we get the next point
which is two comma zero so that's an
x-intercept
the x-intercept is the point
of the graph that touches the x-axis
because on the x-axis y is zero the
y-intercept
touches the y-axis
so this point
is zero negative four
it's the y-intercept because x is zero
and this is the x-intercept because y is
zero
now all you need is two points in order
to graph a line
so we can add more points but
we can just connect these points with a
straight line
so that's how we can graph
this equation in slope intercept form
now let's say we have this one y is
equal to negative three over four
x plus five
how can we graph this equation
feel free to pause the video if you want
to try it
so first let's identify the slope
and the y-intercept
so we can see that the slope
is negative 3 over 4 is the number in
front of x
the y-intercept
is 5.
so first we're gonna
plot the y-intercept
the y-intercept has the point
zero negative five x is zero y is
negative five so it's on the y axis
this is the x axis this is the y axis
so now that we have the first point the
y intercept let's use the slope to get
the next point so starting from the
y-intercept
we have a rise of negative three
and a run
of four
so this is the rise
this is the run
so as we travel down three and over four
it's going to take us to this point
so that is four on the x-axis
two on the y-axis
now all we need is two points
to graph a line so now we can just
draw a line that connects those two
points
and so that's how we can graph that
particular linear function
now let's move on to the next example
so this time we're going to graph a
linear function
in standard form
so we have 3x minus 2y is equal to 6.
so it's an ax
plus by
equals c format
how can we graph a linear equation in
standard form
what do you think we need to do
one of the most simplest techniques that
you can use is to find the x and the y
intercepts
to find the x intercept
replace y with zero
negative two times zero is zero so we
get just three x is equal to six
solving for x we can divide both sides
by three
and so we get x is equal to
six divided by three which is two
so the x intercept
is two comma zero
x is two
y is zero since we replaced y with zero
now let's find the y intercept
to find the y intercept replace x with
zero
three times zero is zero
so we're just going to get negative two
y is equal to six
dividing both sides by negative two
we get that y
is six divided by negative two which is
negative three
so the y intercept is going to be 0
negative 3.
so what we're going to do is we're going
to plot the x intercept
which is
here
it's 2 comma zero
and then let's plot the widest up
the y intercept is zero negative three
so now let's connect
these two points with a straight line
and that's all you need to do in order
to graph a linear equation in standard
form
let's try another example so let's say
we have 4x
plus 3y is equal to 12.
go ahead and graph that linear equation
so let's find the x-intercept
let's replace y
with zero
so we're going to get 4x is equal to 12
and then dividing both sides by four
we get x is 12 over four which is three
so the x intercept is three comma zero
now let's get the whiteness up
so this time let's replace x with zero
four times zero is zero
so we just get three y is equal to
twelve
divide both sides by three
so y is twelve divided by three which is
four
so we get the point zero comma four
so the x-intercept is at three
the y-intercept is at four
and then we just need to
connect those two points
with a straight line
so that's how we can graph
the linear equation in standard form 4x
plus 3y is equal to 12.
now what about an equation
that is in point slope form let's say we
have y minus 3
is equal to 2
times x minus 2.
how can we graph an equation
in that form
feel free to try that problem
so this is in y minus y1 is equal to m
times x minus x1 form
that's the point slope form in that form
we could find the point and the slope
so here's the slope the slope is 2.
now we can also find a point through
which the line passes through and that
point
is x1
y1
so what's x1 and what's y1
notice that these two negative signs are
the same
therefore
1 has to be positive 2
because those negative signs
already there
so x 1 is positive 2
y 1 is 3 without the negative sign
so when you see x minus 2 the point is
going to be 2. change the negative sign
into a positive sign
if you see y minus 3 the y coordinate is
positive 3.
so with this information we can graph it
we have a point and a slope
so let's plot the point two three
so here is two three
the x value is two the y value is three
and then we could use the slope
to get the next point
the slope is two
so we have a rise of two
and a run of one
so we can go up two and over one to get
the next point
so that's going to be three comma five
and we can go backwards
let's say if we go one to the left we
need to go down to
because there's not much space
in the right side of this graph
so that's how we can graph
a linear equation in point slope form
for the sake of practice let's do one
more example
so let's say we have the linear equation
y plus four is equal to negative three
over two
times x plus one
so go ahead and graph that linear
equation
so let's begin by identifying the slope
the slope
is negative three over two
now what's the point here we have x plus
one the x coordinate is going to be
negative one
simply reverse positive one to negative
one
here we have y plus four the y
coordinate will be negative four
so now
we have a point and a slope
that's all we need in order to graph
this function
so the first point is that negative one
negative four which is here
the x value is negative one the y value
is negative four
and then to get the next point
the slope is negative three over two
so we need to
go down three and over two but it looks
like we're out of space
so we're going to go backwards
that is we're going to go up three
and then two to the left
so up three two to the left that still
gives us the same slope
that's a rise of three a run of negative
two
which is still negative three over two
so sometimes you may need to go
backwards like in this problem
so if we go up three and over two we
should be at this point
and this point is at negative three
comma negative one
now at this point we can
go ahead and draw a line
between these two points
so that's a rough sketch
of the graph that corresponds to this
linear equation
now what would you do to graph this
equation
let's say y is equal to 3 how can you
graph that
whenever y is equal to a constant number
what you're going to get is a horizontal
line
in this case a horizontal line
at three
so if we wanted to graph y is equal to
negative two
we would simply draw a horizontal line
at negative two along the y axis
so whenever y is equal to a constant
you're going to get a horizontal line
and the slope of that line is going to
be 0.
now what if we wanted to graph x is
equal to four
in this case we're going to have a
vertical line
at x equal four
so this line will contain all points
with the x coordinate x equals four
the slope of that line
is undefined
if we want to graph x is equal to
negative 3
it's simply going to be a vertical line
touching all points
with the x coordinate negative 3.
so that's how we can graph that
now let's work on some multiple choice
and free response practice problems
that's going to help you to review for
the tests if you're studying for one
number one which of the following graphs
correspond to the equation
y is equal to two x minus three
so this equation is in slope intercept
form
now there's two things we need to focus
on
we need to identify the slope and the
y-intercept
the slope is the number in front of x
so therefore the slope
is equal to 2.
the y-intercept
is the constant that you see
next to the 2x
so the y-intercept which is b
is negative 3.
so let's identify the graph with the
correct y-intercept if we look at answer
choice a
the graph touches the y-axis at positive
three
therefore
answer choice a is not correct
looking at b c and d
the graph touches it at
negative three
so far c b and d are okay
now let's look at the slope
the first thing we want to notice is
that the slope is positive
a positive slope means that the function
is increasing
a negative slope means that it's
decreasing
and for a horizontal line the slope is
zero
so because the slope is positive the
graph should be going up
therefore we can delete d because it's
going down
graph d has a negative slope
now between b and c
what's the difference
well let's look at c
as we travel
one unit to the right
notice that the graph goes up by three
so the slope is three
now let's look at b
as we travel one unit to the right
notice that the graph goes up by two
which gives us a slope of two
so b is the right answer
it has a y-intercept of negative three
and a slope of two
you
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