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Linear Equations - Algebra — Transcript

by The Organic Chemistry Tutor · 3,917 words · 737 segments · language en · Watch on YouTube

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0:00in this video we're going to do a review
0:03of linear equations
0:05it's especially for those of you who
0:06have a test that you're studying for
0:09so
0:10let's begin by writing down some notes
0:13there are three forms in which you can
0:14write a linear equation
0:16the first one
0:18is known as the slope
0:20intercept form
0:26in the slope intercept form the linear
0:28equation is written this way
0:30y
0:31is equal
0:32to mx plus b
0:36m
0:37represents the slope which we'll talk
0:39about later
0:41and b
0:43that represents the y-intercept
0:46we'll also
0:48talk more about that later as well but
0:50for now
0:51you want to write this equation so this
0:53is the slope-intercept form
0:55of a linear equation
1:01now the next form that you want to be
1:02familiar with
1:04is
1:05the standard form
1:08so to write a linear equation in
1:09standard form
1:11this is how it's going to look like
1:15it's a x
1:17plus b y
1:18is equal to c
1:21a b and c are simply coefficients
1:24x and y are the variables
1:26but when written in that form it's
1:28called
1:29the standard form
1:31the next form that you need to be
1:32familiar with is
1:36this one the point-slope form of a
1:39linear equation
1:44it's y minus y one
1:47is equal to m
1:49times x minus x one
1:52as the name implies this equation can
1:54tell you the slope and the point
1:57the slope is the value of m
1:59so whatever number you see here
2:01and that's the slope
2:03the point
2:04is x1 comma y1
2:09now let's talk about the slope
2:14the slope is equal to the rise
2:18divided by the run
2:23so let's say if you have a linear
2:24equation
2:26that is rising
2:27the slope is going to be positive
2:30and let's say you have two points on
2:32this line
2:35now to go from the first point
2:37to the second point
2:39let's say it takes you have to travel
2:41up
2:43by four units
2:45this is not drawn to scale by the way
2:46this is just an illustration so let's
2:48say you travel up four units
2:51and then you travel three units to the
2:53right
2:55so in this case your rise is four
2:58your run is three so rise over run the
3:01slope would be four over three
3:04and because
3:06it's going up the line is going up the
3:08slope is going to be positive
3:11now here's another example
3:14let's say
3:15to go from this point to that point
3:19we need to go down
3:21three units
3:23so the rise is negative because we're
3:24going down so let's say negative three
3:26units
3:27let's say we have a run of positive five
3:30it's positive because we're moving to
3:31the right
3:32the run should always be positive
3:38now for this one the slope is going to
3:39be rise over run
3:41the rise is negative 3 the run is 5 so
3:44it's going to be negative 3 over 5.
3:47so because
3:49the line is going down the slope is
3:51negative
3:52so that's a quick and simple way to
3:54calculate the slope
3:56using the rise over run method
4:05now whenever a line
4:06it goes up at a 45 degree angle
4:09the slope of that line is going to be
4:11one
4:14if it goes up like this it's about one
4:16half
4:18and if it goes up even steeper let's say
4:20like this
4:21this would be a slope of two
4:26so this line is very steep compared to
4:28the other ones
4:30now if the line is horizontal
4:33the slope
4:34is going to be zero
4:37if it goes down at a 45 degree angle
4:39like this
4:40the slope is negative one
4:43here it's about negative a half
4:47and here
4:48negative two
4:51so any time the slope increases i mean
4:54anytime the line increases the slope is
4:56going to be positive
4:59if the graph is going down the slope is
5:01negative
5:03and for any horizontal line the slope is
5:05zero
5:07so if we have a line that's going to the
5:08right or to the left the slope is zero
5:11for a vertical line
5:13the slope is undefined
5:21as the line becomes more vertical the
5:22slope increases
5:25eventually it can go up to infinity
5:27and at some point it will be undefined
5:32so just know that so if you have a
5:34vertical line the slope is undefined for
5:36a horizontal line the slope is zero
5:40now you can calculate the slope of a
5:42line
5:43if you know the two points
5:45so let's say if the first point
5:47is x one comma y one
5:50and the second point
5:51is x2 y2
5:54the slope of the line is going to be
5:56y2
5:58minus y1
5:59divided by
6:01x2
6:03minus x1 so here's an example let's say
6:05the first point
6:07is two comma five
6:09and the second point
6:11is let's say five
6:16fourteen
6:19go ahead and calculate the slope
6:22so y2 is going to be 14
6:26and let's replace y1 with five
6:31x2 is five
6:34x one is two
6:37fourteen minus five is nine
6:40five minus two is three
6:42nine divided by three is three
6:44so the slope
6:46of the line that connects
6:49these two points
6:50is equal to 3.
6:54now let's talk about
6:56x and y-intercepts
7:01what is an x-intercept
7:03and what is
7:05a y-intercept
7:12what do you think
7:14the answer to that question is
7:18an x-intercept is a point
7:22but it's a specific point
7:25the x-intercept is the point
7:27where y is equal to zero
7:30so let's say you have the point three
7:32comma zero
7:34at this point x is three
7:36y is zero
7:38this particular point is an x intercept
7:40because
7:41the y value is zero
7:43so the x-intercept is any value of x
7:46when y is zero
7:48another example of an x-intercept is the
7:50point
7:50negative five zero
7:52so the x-intercept in this case will be
7:55negative five
7:57if we have the point
7:59two comma zero
8:00the x intercept is two
8:04so any point where the y value is zero
8:08the x value is the x intercept
8:11for the y-intercept the situation is
8:13reverse
8:14the y-intercept is the y-coordinate of a
8:17point when x is 0.
8:19so let's say if we have the point
8:220
8:234
8:24the y-intercept is 4.
8:28so remember when dealing with linear
8:30equations the y-intercept is also equal
8:32to b
8:33so we would say that b
8:35is 4.
8:37here's another example of a y-intercept
8:39let's say the point zero negative three
8:43so x is zero y is negative three the y
8:45intercept
8:47is negative three so we can say that b
8:49is equal to negative three
8:51now let's summarize what we've just
8:53learned
8:54the x intercept
8:56is the x coordinate
8:58of a point
8:59that contain a y value of zero
9:02so as we see here
9:04this point has a y value of zero the x
9:06intercept is the x-coordinate of that
9:08point
9:09so x is negative five
9:13the y-intercept
9:15is the y-coordinate of a point that has
9:17an x-value of zero
9:20so for this point
9:23the x value is 0 but the y intercept is
9:26the y coordinate of that point so it's y
9:28equals 4 or b equals 4.
9:32so that's the basics of the x and y
9:34intercepts
9:39so here's an example problem for you
9:41consider
9:42these four points the point two comma
9:44five
9:46negative three comma zero
9:50one comma two
9:53and zero comma six
9:56given these four points identify the x
9:59and y-intercepts
10:03so this point here has a y value of zero
10:07therefore that point represents the
10:09x-intercept
10:11the x-intercept is specifically the
10:13x-coordinate of that point so the
10:15x-intercept
10:16we could say it's x equals negative
10:18three
10:21but it you can also say that the
10:22x-intercept is the point negative three
10:24comma zero you can describe it both ways
10:28the y-intercept is the point where x is
10:31zero so this would be the y intercept
10:35now you can say the y intercept is y
10:37equals six
10:38or you could say it's a b equals six
10:42now the next thing that we need to
10:43review
10:45are parallel lines
10:47and perpendicular lines
10:51so what's the difference between
10:53parallel lines and perpendicular lines
10:55what would you say
11:01and how do their slopes
11:02relate to each other
11:08parallel lines
11:09they travel in the same direction
11:12let's call this line one and line two
11:16let's say the slope of line one has a
11:18value of two
11:20if the slope of line one has a value of
11:22two and if line two is parallel to line
11:25one
11:26the slope of line two will be the same
11:29it will also be two
11:31what you need to know is that parallel
11:33lines they have the same slope
11:36so m1 is going to be equal
11:38to m2
11:42you can also describe the relationship
11:45between two lines that are parallel
11:46using the symbol
11:49if you see this
11:51double vertical line
11:53it means that the two lines are parallel
11:55so this is saying l1 is parallel to l2
11:59but
12:01for linear equations if you have a test
12:02we need to know is that parallel lines
12:05they have the same slope
12:09now let's talk about perpendicular lines
12:11let's say we have this line
12:13which we'll call l1
12:16and then this line l2
12:23perpendicular lines
12:25they intersect at right angles that is
12:28at 90 degrees
12:33the slope of the perpendicular line is
12:36the negative reciprocal of the original
12:37line
12:38so let's say the slope of line one
12:42let's say it's positive
12:44three over four
12:48the slope of line two is going to be the
12:50negative reciprocal so you've got to
12:52change the sign from positive to
12:53negative
12:54and you've got to flip the fraction
12:57so it's going to be negative
12:584 over 3.
13:02so m1 is going to be
13:04negative
13:051 over m2
13:07so that's the relationship between the
13:09slopes
13:10of two perpendicular lines
13:13so we could say l1
13:16is perpendicular this is the symbol for
13:18perpendicular l1 is perpendicular to l2
13:22you can see these two lines meet at
13:23right angles
13:25so that's how you can describe two
13:27perpendicular lines
13:28so remember the slopes
13:30are negative reciprocals of each other
13:33now let's work on some example problems
13:35let's say that line one
13:38is parallel
13:40to line two
13:42and let's say that you're given the
13:43slope of line one
13:45let's say that
13:46the slope of line one is negative three
13:49what is the slope of line two
13:53if they're parallel the slope of line
13:56one is equal to the slope of line two
13:59therefore the slope of line two will be
14:01negative three
14:04now let's say that line one
14:06is perpendicular to line two
14:11and let's say you're given the slope of
14:12line one let's say
14:14it's
14:16negative four
14:17over seven
14:19what is the slope of line two
14:22the slope of line two
14:24is going to be the negative reciprocal
14:27of the slope of line one
14:30so the first thing you need to do is
14:32change the sign from
14:33negative
14:34to positive
14:37and then you need to flip the fraction
14:38from four over seven to seven over four
14:42so that's going to be the slope
14:44of the line that's perpendicular to the
14:46first line
14:47now let's talk about how we can graph
14:49equations
14:51in slope intercept form
14:52so let's say we have the equation y
14:55is equal to 2x minus 4.
14:58how can we graph this equation
15:03so first let's put in some
15:06marks on a graph
15:12feel free to try this example if you
15:13want to
15:21so the first thing we need to identify
15:23is the slope and the y intercept so this
15:26is in
15:27y equals mx plus b form it's in slope
15:31intercept form
15:32so we can see that the slope
15:34is equal to two
15:38and we can see that the y-intercept
15:41is negative four
15:45with this information we have everything
15:46that we need in order to graph this
15:49function
15:50so here's negative four
15:52let's go ahead and plot the y-intercept
15:57and then from the y-intercept we can get
15:59the second point by using the slope
16:02so the slope is 2
16:04which means that it's 2 over 1.
16:07so the rise is two the run is one so to
16:10get the next point
16:12we're going to go
16:13up two units
16:15and then travel one unit to the right
16:18so that will give us the point
16:23one negative two so we have an x value
16:25of one
16:27and a y value of negative two
16:31now let's go up two and over one again
16:35so we get the next point
16:37which is two comma zero so that's an
16:38x-intercept
16:41the x-intercept is the point
16:44of the graph that touches the x-axis
16:47because on the x-axis y is zero the
16:50y-intercept
16:51touches the y-axis
16:54so this point
16:55is zero negative four
16:58it's the y-intercept because x is zero
17:00and this is the x-intercept because y is
17:02zero
17:08now all you need is two points in order
17:10to graph a line
17:12so we can add more points but
17:14we can just connect these points with a
17:15straight line
17:24so that's how we can graph
17:26this equation in slope intercept form
17:31now let's say we have this one y is
17:33equal to negative three over four
17:36x plus five
17:37how can we graph this equation
17:41feel free to pause the video if you want
17:43to try it
17:48so first let's identify the slope
17:51and the y-intercept
17:55so we can see that the slope
17:57is negative 3 over 4 is the number in
17:59front of x
18:05the y-intercept
18:07is 5.
18:21so first we're gonna
18:22plot the y-intercept
18:24the y-intercept has the point
18:26zero negative five x is zero y is
18:29negative five so it's on the y axis
18:33this is the x axis this is the y axis
18:37so now that we have the first point the
18:39y intercept let's use the slope to get
18:41the next point so starting from the
18:43y-intercept
18:44we have a rise of negative three
18:48and a run
18:49of four
18:52so this is the rise
18:56this is the run
19:03so as we travel down three and over four
19:06it's going to take us to this point
19:10so that is four on the x-axis
19:13two on the y-axis
19:17now all we need is two points
19:19to graph a line so now we can just
19:22draw a line that connects those two
19:23points
19:28and so that's how we can graph that
19:29particular linear function
19:31now let's move on to the next example
19:34so this time we're going to graph a
19:35linear function
19:37in standard form
19:38so we have 3x minus 2y is equal to 6.
19:42so it's an ax
19:44plus by
19:45equals c format
19:49how can we graph a linear equation in
19:51standard form
19:52what do you think we need to do
20:02one of the most simplest techniques that
20:04you can use is to find the x and the y
20:06intercepts
20:08to find the x intercept
20:10replace y with zero
20:14negative two times zero is zero so we
20:16get just three x is equal to six
20:19solving for x we can divide both sides
20:21by three
20:23and so we get x is equal to
20:26six divided by three which is two
20:29so the x intercept
20:31is two comma zero
20:33x is two
20:34y is zero since we replaced y with zero
20:40now let's find the y intercept
20:42to find the y intercept replace x with
20:45zero
20:46three times zero is zero
20:48so we're just going to get negative two
20:50y is equal to six
20:52dividing both sides by negative two
20:54we get that y
20:56is six divided by negative two which is
20:58negative three
20:59so the y intercept is going to be 0
21:02negative 3.
21:07so what we're going to do is we're going
21:08to plot the x intercept
21:10which is
21:12here
21:14it's 2 comma zero
21:16and then let's plot the widest up
21:24the y intercept is zero negative three
21:27so now let's connect
21:29these two points with a straight line
21:32and that's all you need to do in order
21:34to graph a linear equation in standard
21:36form
21:38let's try another example so let's say
21:40we have 4x
21:41plus 3y is equal to 12.
21:44go ahead and graph that linear equation
22:00so let's find the x-intercept
22:02let's replace y
22:04with zero
22:07so we're going to get 4x is equal to 12
22:10and then dividing both sides by four
22:14we get x is 12 over four which is three
22:17so the x intercept is three comma zero
22:21now let's get the whiteness up
22:23so this time let's replace x with zero
22:28four times zero is zero
22:30so we just get three y is equal to
22:32twelve
22:34divide both sides by three
22:36so y is twelve divided by three which is
22:39four
22:41so we get the point zero comma four
22:45so the x-intercept is at three
22:47the y-intercept is at four
22:50and then we just need to
22:52connect those two points
22:54with a straight line
22:58so that's how we can graph
23:00the linear equation in standard form 4x
23:03plus 3y is equal to 12.
23:06now what about an equation
23:09that is in point slope form let's say we
23:11have y minus 3
23:12is equal to 2
23:14times x minus 2.
23:18how can we graph an equation
23:20in that form
23:23feel free to try that problem
23:40so this is in y minus y1 is equal to m
23:45times x minus x1 form
23:47that's the point slope form in that form
23:50we could find the point and the slope
23:53so here's the slope the slope is 2.
23:59now we can also find a point through
24:01which the line passes through and that
24:04point
24:05is x1
24:08y1
24:09so what's x1 and what's y1
24:12notice that these two negative signs are
24:14the same
24:15therefore
24:171 has to be positive 2
24:19because those negative signs
24:21already there
24:24so x 1 is positive 2
24:27y 1 is 3 without the negative sign
24:34so when you see x minus 2 the point is
24:36going to be 2. change the negative sign
24:37into a positive sign
24:39if you see y minus 3 the y coordinate is
24:41positive 3.
24:43so with this information we can graph it
24:47we have a point and a slope
24:50so let's plot the point two three
24:52so here is two three
24:56the x value is two the y value is three
25:00and then we could use the slope
25:01to get the next point
25:04the slope is two
25:06so we have a rise of two
25:09and a run of one
25:12so we can go up two and over one to get
25:14the next point
25:18so that's going to be three comma five
25:22and we can go backwards
25:24let's say if we go one to the left we
25:26need to go down to
25:28because there's not much space
25:31in the right side of this graph
25:37so that's how we can graph
25:39a linear equation in point slope form
25:43for the sake of practice let's do one
25:44more example
25:46so let's say we have the linear equation
25:48y plus four is equal to negative three
25:51over two
25:52times x plus one
25:54so go ahead and graph that linear
25:56equation
26:07so let's begin by identifying the slope
26:11the slope
26:12is negative three over two
26:18now what's the point here we have x plus
26:20one the x coordinate is going to be
26:22negative one
26:23simply reverse positive one to negative
26:25one
26:26here we have y plus four the y
26:28coordinate will be negative four
26:30so now
26:31we have a point and a slope
26:34that's all we need in order to graph
26:36this function
27:02so the first point is that negative one
27:05negative four which is here
27:08the x value is negative one the y value
27:10is negative four
27:12and then to get the next point
27:15the slope is negative three over two
27:19so we need to
27:21go down three and over two but it looks
27:24like we're out of space
27:26so we're going to go backwards
27:28that is we're going to go up three
27:30and then two to the left
27:33so up three two to the left that still
27:36gives us the same slope
27:37that's a rise of three a run of negative
27:39two
27:40which is still negative three over two
27:43so sometimes you may need to go
27:44backwards like in this problem
27:48so if we go up three and over two we
27:50should be at this point
27:52and this point is at negative three
27:55comma negative one
27:58now at this point we can
28:00go ahead and draw a line
28:02between these two points
28:05so that's a rough sketch
28:07of the graph that corresponds to this
28:08linear equation
28:13now what would you do to graph this
28:15equation
28:16let's say y is equal to 3 how can you
28:19graph that
28:21whenever y is equal to a constant number
28:24what you're going to get is a horizontal
28:25line
28:26in this case a horizontal line
28:29at three
28:31so if we wanted to graph y is equal to
28:33negative two
28:35we would simply draw a horizontal line
28:38at negative two along the y axis
28:44so whenever y is equal to a constant
28:47you're going to get a horizontal line
28:49and the slope of that line is going to
28:51be 0.
28:54now what if we wanted to graph x is
28:56equal to four
28:58in this case we're going to have a
28:59vertical line
29:01at x equal four
29:03so this line will contain all points
29:05with the x coordinate x equals four
29:08the slope of that line
29:10is undefined
29:14if we want to graph x is equal to
29:16negative 3
29:18it's simply going to be a vertical line
29:20touching all points
29:22with the x coordinate negative 3.
29:26so that's how we can graph that
29:30now let's work on some multiple choice
29:31and free response practice problems
29:33that's going to help you to review for
29:36the tests if you're studying for one
29:38number one which of the following graphs
29:41correspond to the equation
29:43y is equal to two x minus three
29:46so this equation is in slope intercept
29:48form
29:50now there's two things we need to focus
29:51on
29:52we need to identify the slope and the
29:54y-intercept
29:56the slope is the number in front of x
29:58so therefore the slope
30:01is equal to 2.
30:03the y-intercept
30:05is the constant that you see
30:08next to the 2x
30:10so the y-intercept which is b
30:12is negative 3.
30:14so let's identify the graph with the
30:16correct y-intercept if we look at answer
30:18choice a
30:19the graph touches the y-axis at positive
30:21three
30:22therefore
30:23answer choice a is not correct
30:27looking at b c and d
30:29the graph touches it at
30:31negative three
30:35so far c b and d are okay
30:38now let's look at the slope
30:39the first thing we want to notice is
30:41that the slope is positive
30:44a positive slope means that the function
30:45is increasing
30:48a negative slope means that it's
30:50decreasing
30:51and for a horizontal line the slope is
30:53zero
30:55so because the slope is positive the
30:57graph should be going up
30:58therefore we can delete d because it's
31:01going down
31:05graph d has a negative slope
31:08now between b and c
31:09what's the difference
31:11well let's look at c
31:13as we travel
31:15one unit to the right
31:17notice that the graph goes up by three
31:20so the slope is three
31:26now let's look at b
31:28as we travel one unit to the right
31:31notice that the graph goes up by two
31:33which gives us a slope of two
31:36so b is the right answer
31:39it has a y-intercept of negative three
31:40and a slope of two
32:04you

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