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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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• 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? • Sal is now plotting points that lie on the line defined by the second equation in the system of equations. the relation he is graphing is 6x - 6y = -6. One way to plot a line is to plot any two points that are on the line, and for an equation in standard from like this one, two easy points to find at the x and y intercepts--the values where x=0 and y=0. To find the x intercept, plug y=0 into 6x-6y=-6 and you get 6x-6(0)=-6 which simplifies to 6x=-6 or x=-1. Similarly, to find the y-intercept, let x=0. you get 6(0)-6y=6 which simplifies to -6y=-6 or y=1. So the points (-1,0) and (0,1) are on our line.

You can see a video explaining the process of finding intercepts

• I didn't understand the concept • How much of a lesson should I get done each day on khan academy? • 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? • 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.
• I am so confused. Next time, could you explain slower?
(1 vote) • So how do you plot the dot when, for example, x=2/3? • So, how would you plot something like
7x−y=7

x+2y=6
​ I cannot figure out how to plot it. • 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.
• I don't understand on how he found not the y-intercept, but the other part on graphing. • 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? • 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.
-20x+12y= -24
5x-3y=6

what do i do, because i can't make some of them 0 and get an answer • This is an interesting case of a system of linear equations, because it doesn't result in one unique solution. Using the method from the video above, we can attempt to solve this system graphically. In the first equation:
when x=0:
12y=-24
y=-2 --> (0,-2) is the y-intercept

when y=0:
-20x = -24
x = 1.2 --> (1.2, 0) is the x-intercept

Connecting these dots and extending them would create a line that has the solutions for this equation. Now we just need to find where the line of solutions for the second equation intersects with the first equation. For the second equation:
when x=0:
-3y=6
y=-2 --> (0, -2) is the y-intercept

when y=0:
5x=6
x=6/5=1.2 --> (1.2, 0) is the x-intercept

Notice something odd? When you connect the dots for both of these equations, you get two equal lines laying on top of each other. This means that every point along that line is a solution to this problem. Therefore, there will be an infinite number of solutions. To answer the question we could write the equation of this solution line in slope-intercept form to represent all the values of x and y that would make this system true.

Taking a look at the line created before, b= -2 (y-intercept). To find slope, (use the points found from before) we can use m = (y2-y1)/(x2-x1) = (0- -2)/(1.2-0) = 5/3. Therefore, in slope-intercept form the line that contains all of the solutions to this problem is:
y=mx+b
y= (5/3)x - 2