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Systems of equations with graphing: exact & approximate solutions

Sal solves a system of two linear equations in standard form, and then approximates the solution of a system whose solution isn't clearly visible.

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  • duskpin ultimate style avatar for user ddonalson101
    At , Khan says that when y is zero, x is negative one. I don't get how he got that. Can anyone explain this in another way?
    (12 votes)
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  • male robot donald style avatar for user Mary Silva
    So,like, I don't get this. For example, when Sal says that "When x is equal to zero, y would be equal to negative three" what does that mean?
    (0 votes)
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    • stelly blue style avatar for user Kim Seidel
      As Sal states - He is picking different values of X and then calculating Y using one of the equations. In the one you referenced, Sal is using the first equation: -x-3y=9. If you use x=0, the equation becomes: -0-3y=9, then solve for Y.
      -3y=9
      y=-3
      This creates one point for graphing the first line. The point is (0, -3). Sal repeats this process using other values of X to find 2 points for each line.

      Hope this helps.
      (9 votes)
  • hopper happy style avatar for user JennaSawyer
    I don't understand on how he found not the y-intercept, but the other part on graphing.
    (3 votes)
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  • starky tree style avatar for user Lola0008
    So how do you plot the dot when, for example, x=2/3?
    (3 votes)
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    • stelly blue style avatar for user Kim Seidel
      Fractions sit in between the integers on a number line.
      For x=2/3, it is located between 0 and 1. Split the space between 0 and 1 into 3 equal sections. 2/3 is 2 of those sections from 0.

      Search for the lesson on "fractions on a number line" for more details.
      (1 vote)
  • blobby green style avatar for user CarlosC
    So, how would you plot something like
    7x−y=7

    x+2y=6
    ​ I cannot figure out how to plot it.
    (2 votes)
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    • stelly blue style avatar for user Kim Seidel
      You have a couple of options:
      1) You can convert each equation to slope-intercept form, then graph using the y-intercept and the slope.

      2) You can calculate 2 points for each line. Once you have 2 points for the line, you can draw the line. To find a point, pick a value for X or Y and put it into the equation. Then, calculate the other variable. For example: if y=0
      7x-0=7
      7x=7
      x=1
      You now have the point (1,0) that can be graphed.

      Hope this helps.
      (3 votes)
  • starky ultimate style avatar for user DuncanR
    How much of a lesson should I get done each day on khan academy?
    (2 votes)
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  • aqualine seed style avatar for user Sam Horak-vik
    I didn't understand the concept
    (4 votes)
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    • aqualine ultimate style avatar for user spy
      Pretty much, the concept is to replace the x and y values with simple values and graph the result. In the case of a fraction like 1/3X, the most simple value would probably be 3 as it would remove the fraction. The X and Y values are not set in stone for substitution.
      (0 votes)
  • leaf green style avatar for user Yaw Opoku-Afriyie
    It seems like I have everything else down,but think I'm graphing wrong. I already found my x and y values. I'm confused about this question 2x+3y=6, -4x+3y=12 I'm not good at "graphing" systems of equations yet. Graphing is where I struggle. May someone please walk me through to get the correct answer to this problem? Please and thank you. This question is not in the video by the way it is a practice problem.
    (2 votes)
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    • hopper cool style avatar for user Eric5566
      OK, so you have found your x and y values. You can graph 2x+3y=6, -4x+3y=12. So for 2x+3y=6, your x might be 0 and your y might be 2. So to plot these points on the graph by finding 0 on the x-axis (also called the horizontal line) and you find 2 on the y-axis (also called vertical line). There go, you have your first point (0,2). Continue the rest to form a line. You only need 2 points to make a line, but if the somebody says to use the whole graph, you might as well graph all the valid points.
      (1 vote)
  • hopper cool style avatar for user Aytabrizi🌹
    I am so confused. Next time, could you explain slower?
    (0 votes)
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    • stelly blue style avatar for user Kim Seidel
      1) You can slow down the video yourself. Click on the gear symbol in the lower right of the video window and adjust the video speed.
      2) Use the pause button as soon as you start to get confused. Try reviewing the transcript to see if that helps eliminate your confusion.
      3) You can watch the video as many times as you need to.
      (7 votes)
  • primosaur ultimate style avatar for user Steve
    Okay...I'm thoroughly confused...instead of taking each equation, and making the X =0 ...then the Y=0... can't we just re-arrange the equation to make it in y=mx+b format? then graph them? What's the purpose or need to set x=0 and then y to 0?
    (1 vote)
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    • stelly blue style avatar for user Kim Seidel
      There are multiple ways to graph an equation.
      -- The video is using the intercepts method -- you find the X and Y intercepts and graph those 2 points, then draw the line.
      -- You want to use the slope-intercept form of the equation to graph using the y-intercept and the slope.
      -- You could also find 2 random points on the line by picking values for either X or Y and solving for the other variable.

      All these methods are acceptable. So, if you prefer to graph using the slope intercept, do it. The only time this wouldn't be acceptable is if your teacher or a particular problem told you to use a different method.
      (2 votes)

Video transcript

- The following two equations form a linear system. This is one equation; it has X and Y so it's gonna define a line. And then I have another equation that involves X and Y, so it's gonna define another line. It says: "Graph the system of equations "and find its solution." So we're gonna try to find it visually. So let's graph this first one. To graph this line, I have the little graphing tool here. Notice if I can figure out two points, I can move those points around and it's going to define our line for us. I'm gonna pick two X values and figure out the corresponding Y values and then graph the line. So let's see how I could do this. So let's see; an easy one is what happens when X is equal to zero? Well if X is equal to zero, everything I just shaded goes away and we're left with -3y is equal to nine. So -3y equals nine. Y would be negative three. So when X is equal to zero, Y would be negative three. So let me graph that. When X is equal to zero, X is zero, Y is negative three. Now another easy point actually instead of trying another X value, let's think about when Y is equal to zero 'cause these equations are in a standard form so it's easy to just test. Well what are the X and Y intercepts? So when Y is equal to zero, this term goes away, and you have negative X is equal to nine, or X would be equal to negative nine. So when Y is zero, X is negative nine. So when Y is zero, X is negative nine, or when X is negative nine, Y is zero. So I've just plotted this first equation. So now let's do the second one. We'll do the same thing. What happens when X is equal to zero? When X is equal to zero, so this is going to be our Y intercept now. When X is equal to zero, -6y is equal to negative six. Well Y would have to be equal to one. So when X is zero, Y is equal to one. So when X is zero, Y is equal to one. Get one more point here. When Y is zero, when this term is zero, Y being zero would make this entire term zero, then 6x is equal to negative six or X is equal to negative one. So when Y is zero, X is negative one or when X is negative one, Y is zero. When X is negative one, Y is zero. And so just like that, I've plotted the two lines. And the solution to the system are the X and Y values that satisfy both equations; and if they satisfy both equations, that means they sit on both lines. And so in order to be on both lines, they're going to be at the point of intersection. And I see this point of intersection right over here, it looks pretty clear that this is the point X is equal to negative three and Y is equal to negative two. So it's the point negative three comma negative two. So let me write that down. Negative three comma negative two. And then I could check my answer; got it right. Let's do another. Let's do another one of these. Maybe of a different type. So over here it says: "A system of two linear equations "is graphed below. "Approximate the solution of the system." Alright so here I just have to just look at this carefully and think about where this point is. So let's think about first its X value. So its X value, it's about right there in terms of its X value. It looks like, so this is negative one. This is negative two, so negative 1.5 is gonna be right over here. It's a little bit to the left of negative 1.5, so it's even more negative, I would say negative 1.6. And I'm approximating it, negative 1.6. Hopefully it has a little leeway in how it checks the answer. What about the Y value? So if I look at the Y value here, it looks like it's a little less than one and a half. One and a half would be halfway between one and two. It looks like it's a little less than halfway between one and two, so I'd give it 1.4, positive 1.4. And let's check the answer, see how we're doing. Yep, we got it right. Let's actually just do one more for good measure. So this is another system. They've just written the equations in more of our slope intercept form. So let's see, Y is equal to negative seven, X plus three. When X is equal to zero, we have our Y intercept. Y is equal to three. So when X is equal to zero, Y is equal to three. And then we see that our slope is negative seven. When you increase X by one, you decrease Y by seven. So when you increase X by one, you decrease Y by one, two, three, four, five, six, and seven. When X goes from zero to one, Y went from three to negative four, it went down by seven, so that's that first one. Now the second one: our Y intercept. When X is equal to zero, Y is negative three, so let me graph that. When X is zero, Y is equal to negative three. And then its slope is negative one. When X increases by one, Y decreases by one. So the slope here is negative one. So when X increases by one, Y decreases by one. And there you have it. You have your point of intersection. You have the X-Y pair that satisfies both equations. That is the point of intersection. It's gonna sit on both lines which is why it's the point of intersection. And that's the point X equals one, Y is equal to negative four. So you have X equals one and Y is equal to negative four. And I can check my answer and we got it right.