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### Course: 8th grade (Illustrative Mathematics)>Unit 4

Lesson 11: Lesson 13: Solving systems of equations

# Number of solutions to a system of equations graphically

Sal determines how many solutions the following system of equations has by considering its graph: 10x-2y=4 and 10x-2y=16. Created by Sal Khan.

## Want to join the conversation?

• Possibly not the right place to ask this, but - at , what's Arbegla? Some manner of American cultural reference, I'm assuming.
• Along with being algebra spelled backwards, it is also a reference to his previous videos with a character named Arbegla who was the king's top advisor and party planner: http://www.khanacademy.org/math/cc-eighth-grade-math/cc-8th-systems-topic/cc-8th-systems-overview/v/how-many-bags-of-potato-chips-do-people-eat
Arbegla was the person who was creating problems for the protagonist in Sal's fantasy story to solve, in hopes that he would fail. So when Sal says "so we don't get stumped by the Arbeglas in our life" he means so that we are able to solve problems, through the use of Algebra, that others may ask us in hopes that we don't succeed.
• I'm sorry, I still don't get how Sal solved the problem around .
They are "fundamentally different ratios"...what does that mean?
• He's comparing the "5 to 1" and "4 to 1" ratios of y to x, and saying that they have different slopes. Therefore, the two lines must intersect somewhere at one point.

If you've watched enough videos on here, you'll notice that Sal frequently (over)uses the word "fundamentally," to just mean "certainly" or "definitely." He didn't mean anything special by the use of the word "fundamentally" here.
• So two linear equations will ALWAYS intersect at one point if thier slope is different? I don't really understand what Sal said at . Please help. I will upvote you comment if you help me!
• Correct! If two linear equations have different slopes, they will ALWAYS intersect. Even if their slopes are different by a very small margin and lines themselves are far apart - they will still intersect. These lines extend into infinity and so when the slopes are different, lines will eventually meet, maybe somewhere very high on the graph or somewhere very low but they WILL intersect.

Linear Equations with the same slope are parallel lines and will NEVER intersect, no matter how far they reach into infinity. They will always run parallel to one another.

Linear Equations with the same slope AND same y-intercept (x=0) is one line running on top of another line, into infinity. Any (x,y) point on one line will also satisfy equation for the other line - because both lines are identical - to infinity. Thus infinite number of solutions.

TLDR: Unless lines are parallel (same slope) they will ALWAYS have one solution (intersect).
• what and why is math so mathy
• I always wonder this
(1 vote)
• Why is Sals Khan Academy verison different from ours?
• That is what KhanAcademy looked like several years ago. They didn't redo all the videos after the system changed. Instead, they invested their time on creating new videos with new content to help people learn.
• the heck are these intros
• What do you mean by that?
• At , what if it is a curved line
• I don't think there are curved lines in this.
• At , if the line are exactly the same, what is the point of drawing/graphing two of them?
• It was just to show people who learn this visually.
• How would one solve an equation with no y in it for example:
x=3
y=2x=7
or
x=3.14
x= -3/2
(1 vote)
• If there is no y, or any second variable, then it would just appear as a straight line that crosses the x axis at whatever constant is in the equation. Therefore, two lines that only have an x and no y would either never intersect or they would completely overlap.