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INSTRUCTORSCarleen EatonGrant Fraser
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For more information, please see full course syllabus of Algebra 2
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Lecture Comments (7)

1 answer

Last reply by: Dr Carleen Eaton
Wed Dec 28, 2011 9:04 PM

Post by Jonathan Taylor on December 26, 2011

Dr.Carleen I'm confused how come u did not use 2x+4y=

2 answers

Last reply by: Dr Carleen Eaton
Mon Aug 22, 2011 11:50 PM

Post by William Terry on August 22, 2011

In example III, why didn't you divide both sides of the equation by 12?

1 answer

Last reply by: Sayaka Carpenter
Sat Jul 14, 2012 9:52 AM

Post by Victoria Jobst on May 28, 2011

On Example 3, I saw that you multiplied (y/6) by 3 to get rid of the x/3 on the other side of the equation, and you got (3y/6). Why wasn't it (3y/18)?

Linear Equations

  • A linear equation can be written in the form ax + by = c, where a, b, and c are real numbers. This is called the standard form of the equation.
  • A linear function can be written in the form f(x) = mx + b where m and b are real numbers.
  • The graph of a linear equation or linear function is a straight line.
  • The x coordinate of the point at which the graph crosses the x axis is called the x-intercept. The y-intercept is defined similarly.

Linear Equations

Is the function linear? 4x2 − 3y = 3
No, linear equations do not have a 2nd degree, as is the case in this problem.
Is the function linear? [(4x2)/x] − 3y = 3
  • Try to write the function in standard form ax + by = c
  • you can eliminate one x from the numerator and denominator in [(4x2)/x] − 3y = 3
  • [4x*x/x] − 3y = 3
  • 4x − 3y = 3
After the x is eliminated, the function is a linear equation.
Is the function linear? [(3x2y + 5xy2)/xy] = [5xy/xy]
  • Try to write the function in standard form ax + by = c
  • Notice that the left side of the function can be broken down into two
  • [(3x2y)/xy] + [(5xy2)/xy] = [5xy/xy]
  • Notice how you can eliminate an xy from each term
  • [3xxy/xy] + [5xyy/xy] = [5xy/xy]
  • Simplify
  • 3x + 5y = 5
It is a linear function.
Write in standard form: [y/2] = [x/2] − 10
  • Recall that standard form is ax + by = c. Get rid of the fractions by multiplying by the number in the
  • denominator, 2.
  • 2( [y/2] = [x/2] − 10 )
  • Multiply using the distributive property
  • [(2)y/2] = [(2)x/2] − (2)10
  • Simplify
  • y = x − 20
  • Subtract x from both sides
  • − x + y = − 20
  • Multiply the entire equation by − 1 to make x positive
  • − 1( − x + y = − 20 )
  • x − y = 20
  • Write in standard form: 0 = [3y/2] + [2x/3] − 12
  • Recall that standard form is ax + by = c. Get rid of the fractions by multiplying by the product of
  • the numbers in the denominator:: 2*3 =
  • 2*3( 0 = [3y/2] + [2x/3] − 12 )
  • Multiply using the distributive property
  • (2*3)0 = [(2*3)3y/2] + [(2*3)2x/3] − (2*3)12
  • Simplify
  • 0 = [(\cancel2*3)3y/\cancel2] + [(2*\cancel3)2x/\cancel3] − (2*3)12
  • 0 = 9y + 4x − 72
  • add 72 to both sides
  • 72 = 9y + 4x
4x + 9y = 72
Write in standard form: [4x/5] = [2y/2] + 2
  • Recall that standard form is ax + by = c. Get rid of the fractions by multiplying by the product of
  • the numbers in the denominator:: 5*2 =
  • 5*2( [4x/5] = [2y/2] + 2 )
  • Multiply using the distributive property
  • [(5*2)4x/5] = [( 5*2 )2y/2] + (5*2)2
  • Simplify
  • [(\cancel5*2)4x/\cancel5] = [( 5*\cancel2 )2y/\cancel2] + (5*2)2
  • 8x = 10y + 20
  • Subtract 10y from both sides.
8x − 10y = 20
Write in standard form: [5y/3] = [6x/5] − 1
  • Recall that standard form is ax + by = c. Get rid of the fractions by multiplying by the product of
  • the numbers in the denominator:: 3*5 =
  • ( 3*5 )( [5y/3] = [6x/5] − 1 )
  • Multiply using the distributive property
  • [( 3*5 )5y/3] = [( 3*5 )6x/5] − ( 3*5 )1
  • Simplify
  • [(\cancel3*5)5y/\cancel3] = [( 3*\cancel5 )6x/\cancel5] − ( 3*5 )1
  • 25y = 30x − 15
  • Subtract 10y from both sides.
8x − 10y = 20
Find the intercepts and graph the equation: 2x + 6y = 12
  • Recall that the x - intercep happens when y = 0 and the y - intercep happens when x = 0.
  • For that reason, when looking for either one, you set the other equal to zero.
  • Complete the table to locate the x - and y - intercept
  •   X-Intercept, (x,0) and set y = 0 Y-Intercept, (0,y) and Set x=0
    2x + 6y=12    
  •   X-Intercept, (x,0) and set y = 0 Y-Intercept, (0,y) and Set x=0
      2x +6(0) = 12 2x + 6y = 12
      2x = 12 2(0) + 6y = 12
    2x + 6y=12 x = 6 6y = 12
      (6,0) y = 2
        (0,2)
  • Plot the points and draw the graph.
Find the intercepts and graph the equation: − 2x + 4y = − 8
  • Recall that the x - intercep happens when y = 0 and the y - intercep happens when x = 0.
  • For that reason, when looking for either one, you set the other equal to zero.
  • Complete the table to locate the x - and y - intercept
  •   X-Intercept, (x,0) and set y = 0 Y-Intercept, (0,y) and Set x=0
    -2x + 4y = -8    
  •   X-Intercept, (x,0) and set y = 0 Y-Intercept, (0,y) and Set x=0
      -2x + 4(0) = -8 -2x + 4y = -8
      -2x = -8 -2(00 + 4y = -8
    -2x + 4y = -8 x = 4 4y = -8
      (4,0) y = -2
        (0,-2)
  • Plot the points and draw the graph.
Find the intercepts and graph the equation: 3x − 5y = − 11
  • Recall that the x - intercep happens when y = 0 and the y - intercep happens when x = 0.
  • For that reason, when looking for either one, you set the other equal to zero.
  • Complete the table to locate the x - and y - intercept
  •   X-Intercept, (x,0) and set y = 0 Y-Intercept, (0,y) and Set x=0
    3x - 5yy = -11    
  •   X-Intercept, (x,0) and set y = 0 Y-Intercept, (0,y) and Set x=0
      3x -5(0) = -11 3(0) - 5y = -11
      3x = -11 -5y = -11
    3x - 5yy = -11 x = [(−11)/3] y = [11/5]
      ([(−11)/3],0) (0,[11/5]
       
  • Plot the points and draw the graph.

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

Answer

Linear Equations

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

  • Intro 0:00
  • Linear Equations and Functions 0:07
    • Linear Equation
    • Example: Linear Equation
    • Example: Linear Function
  • Standard Form 2:02
    • Integer Constants with No Common Factor
    • Example: Standard Form
  • Graphing with Intercepts 4:05
    • X-Intercept and Y-Intercept
    • Example: Intercepts
    • Example: Graphing
  • Example 1: Linear Function 7:53
  • Example 2: Linear Function 9:10
  • Example 3: Standard Form 10:04
  • Example 4: Graph with Intercepts 12:25

Transcription: Linear Equations

Welcome to Educator.com.0000

In today's lesson, we will be covering linear equations.0002

Again, this is some review from Algebra I, which we will discuss here.0008

And if you need further detail, go back to the Algebra I lectures and check those out for this subject.0011

So, a linear equation is an equation of the form ax + by = c, where, a, b, and c are constants.0018

For example, 2x + 5y = 3, and here a = 2, b = 5, and c = 3.0028

Another example could be x - 7y = 4; here, even though it is not written, there actually is a 1 in front of the x.0041

Just by convention, we don't write it out when the coefficient is 1.0052

So, a = 1, b = -7, and c = 4.0055

We will talk a little bit more about what this form is.0062

A linear equation represents a function, which is known as a linear function; and that can be written in the form f(x) = mx + b.0067

And this is a very useful form of the equation that will be discussed in another segment.0078

And an example of this would be something such as f(x) = 4x + 2; you may also see it written as y = 4x + 2.0084

And here, m = 4; b = 2; and these two numbers represent certain elements that we will talk about in a few minutes.0107

OK, so the first form of the linear equation that we talked about just a minute ago is also called standard form.0118

So, a linear equation is in standard form if it is written in the form ax + by = c,0129

where a, b, and c are integer constants (and this is important) with no common factor.0135

If a common factor remains, it is not in standard form.0140

For example, if I have an equation 9x + 4y = 10, this is in standard form.0144

I have ax (9 is a), plus by (b is 4), equals c (c is 10), and there is no common factor.0156

Consider another possible equation: 6x + 8y = 12; this is not in standard form, because 6, 8, and 12 have a common factor.0168

What I need to do is divide both sides of this equation by 2 to get 3x + 4y = 6; now, it is in standard form.0189

Another example would be if I had 7y + 5 = -x; again, it is not in standard form.0207

I can add x to both sides to get x + 7y + 5 = 0, and then subtract 5 from both sides to get x + 7y = -5; this is standard form.0218

We can graph linear equations using the intercept method.0246

Recall that the point where a line crosses the x-axis is its x-intercept.0250

And the point where the graph crosses the y-axis is the y-intercept.0259

For example, if I had a line such as this, first the x-intercept is right here; the x-intercept is -2.0265

And at the x-intercept, y is always going to be 0; so, the coordinate pair would be (-2,0), because it crosses the x-axis, and so y is 0.0283

The y-intercept is right here: y equals 4, and at this point, x is going to equal 0, so (0,4) is the y-intercept.0294

And we can use this knowledge in order to graph a linear equation.0308

For example, if I have a linear equation 2x + 3y = 6, all I need in order to plot this are two points on the line.0314

Now, I can easily get two points by finding the x- and y-intercepts.0328

I know that, with the y-intercept, x will be 0; so if I let x equal 0, that is going to give me 0 times 2 is 0,0334

so that will be 3y = 6; 6 divided by 3--y equals 2.0343

OK, to find my second point, I am going to find the y-intercept (this is actually the x-intercept right here)...0.0354

Excuse me, this is the y-intercept right here; now I am looking for the x-intercept.0368

The x-intercept is the point at which y equals 0, and the line of the graph crosses the x-axis.0375

So, this time, I am going to let y equal 0; so I am going to have 2x, plus 3 times 0, equals 6, or 2x + 0 = 6.0383

So then, I end up with x = 3; this is the x-intercept.0395

Now, I have two points; I can plot the line.0400

At the y-intercept, x is 0; y is 2; the y-intercept is right here, at (0,2); the x-intercept is right here at (3,0).0405

Now, I have two points; and when I have two points, that means I can connect them to form a line.0422

And I did that by finding my x-intercept at (3,0) and my y-intercept at (0,2).0431

The technique is to find the x- and y- intercepts by letting x equal 0, and putting that into the equation,0449

and solving for y to get the y-intercept, and letting y equal 00458

to find the x-intercept, and then using those two points to plot the line.0467

In the first example, is the function linear? Well, recall that a linear function can be written in standard form.0475

And standard form is ax + by = c: I look at what I have here, and I have ax (so it seems like I am going along OK),0482

minus by (or that would be the same as plus -by, so that is fine), equals c; but then I have this over here.0492

And this is not part of standard form.0500

And you could attempt to get rid of that; maybe you say, "OK, I will multiply both sides by x to get rid of it."0502

But then, see what happens: all right, let's see what happens if I try to multiply both sides by x to get rid of this x in the denominator.0508

OK, I end up with 2x2 - 3xy = 4x + 1; so I am no better off.0518

I still don't have it in standard form; there is no way for me to get this in standard form; therefore, this is not a linear function.0530

It is not in standard form; I can't get it into standard form.0546

OK, is this function linear? Well, recall that we talked earlier about the form g(x) = mx + b.0550

And remember, g(x) is the same idea as f(x): you can use different letters, so it is still just that we are looking for this form for a linear function.0566

Well, I am looking at this, and I have what looks like an mx and a b, but the problem is that I also have this.0580

And this is not a part of the form for a linear function--this x2 term--therefore, this is not a linear function.0588

Once again, I can't get it into the right form for a linear function.0598

OK, in Example 3, we are asked to write the equation in standard form.0604

And recall that standard form is ax + by = c.0608

And I am dealing with some fractions, and I need to get rid of those in order to get this in standard form.0613

I can do that by multiplying first both sides of the equation by 3; this is going to give me 3y/6 =...here, the 3's will cancel out; -7 times 3--that is -21.0620

I can simplify this to y/2 = x - 21; so, I am still left with a fraction.0637

I am going to multiply both sides of the equation by 2 to eliminate that fraction.0644

The 2's cancel out, so I have y = 2x...-21 times 2; that is -42.0652

Next, I am going to subtract 2x from both sides to get the x on the left side of the equation.0665

And just switching around the order of the x and y to match this: -2x + y = -42.0677

And you could stop here, or you could multiply both sides by -1; and that is more conventional, to have the a term be positive.0688

So, that is going to leave me with 2x - y = 42.0698

I started out with this equation, and I wanted to get it into this form.0707

And I could achieve that by first multiplying by 3 to get rid of this fraction, then (I ended up with this)0711

multiplying by 2 to get rid of the y fraction, then simply subtracting 2x from both sides to get the 2x (the ax)0721

on the left side of the equation, and finally just a little more work to clean this up, multiplying both sides by -1.0730

This is the equation in standard form, where a equals 2, b equals -1, and c equals 42.0738

Example 4: Find the intercepts and graph the equation: 4x - 5y = 20.0748

We already talked about how, with the x- and y-intercepts, you can find those, and then you will have two points, and you can graph an equation.0756

So, first, in order to find the y-intercept, I am going to let x equal 0; this will give me the y-intercept, the point at which the line crosses the y-axis.0766

So, to find the y-intercept, I am going to substitute 0 for x.0779

This is going to give me 0 - 5y = 20, or -5y = 20; divide both sides by -5 to get y = -4.0788

OK, next I want to find the x-intercept: to find that, I am going to let y equal 0.0801

I am going to go back and do this again, this time letting y equal 0.0810

This is for the y-intercept, and this is for the x-intercept.0821

This is going to give me 4x - 0 = 20, or 4x = 20, which is x = 5.0824

Now that I have my x- and y-intercepts, I actually have two points on the line.0834

My first point is (0,-4), the y-intercept; my second point is at (5,0), and that is the x-intercept.0839

OK, connecting these two points gives me the graph of this equation.0859

We use the intercept method in order to graph, finding the x- and y-intercepts,0869

plotting those out on the coordinate plane, and then using those to form a line.0874

That concludes this lesson for Educator.com; and I will see you here again soon.0880