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Systems of equations number of solutions: fruit prices (1 of 2)

Sal gives an example of a system of equations that has no solution! Created by Sal Khan.

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• So In order for a system to have a solution, when graphed they have to intersect at some point?
• Yes. And if they intersect at more than one point it has infinitely many solutions (this only applies to linear systems of equations though -- quadratic's can intersect twice in a system).

edit: or directly overlap, hence also infinitely many solutions
• Arbegla has been stealing from the king! Stealing from the people!
• why is he soo good at drawing?
• He practiced?
• How come the lines of the two equations did not intercepts in (-2a) value ?
(Why did they went in parallel though they share the same (x) value?)

Anyone to clear this confusion please.
• -2 is the slope of both lines, and a is the variable that is the input (x is just the most common variable to use for input). For any value of a, the second equation will have a y value that is two units above the first equation. I have no clue what you are talking about sharing the same x value, Most functions including linear functions share common x values (from negative infinity to positive infinity), but these two equations have no common (x,y) point. Lines with the same slope and different intercepts will always be parallel.
• silly arbeglas
• I thought the green text up top said 31 lbs of apples and 61 lbs of bananas. Let me tell you, I got a very strange answer.
• Why won't we get something incorrect when using substitution in this example though?
2a + b = 3
2a = 3 - b
a = 1.5 - 0.5b

6a + 3b = 15
6(1.5 + 0.5b) + 3b = 15
9 + 3b + 3b = 15
9 + 6b = 15
6b = 6
b = 1

a = 1.5 - 0.5*1
a = 1

2*1 + 1*1 = 3
And this seems to be correct, even though it clearly isn't when we use all the other methods.
Why?
(1 vote)
• You have a sign error. When you substituted a = 1.5 - 0.5b, you switched it to a = 1.5 + 0.5b
Here's what happens if you use the correct sign.
6(1.5 - 0.5b) + 3b = 15
9 - 3b + 3b = 15
9 = 15
You now have a contradiction just like Sal has in the video.
• Aren't there any systems with 4 variables or more?
• Yes, there are. Actually, there are infinitely many and you can make infinitely many variables too.
• Please answer as quickly as possible, in the slope intercept form why do we not use the x intercept and only the y intercept in the equation??
• For an equation of the form y = mx + b, b is the y-intercept because when x = 0, y = m*0 + b = 0 + b = b.

If you want the equation to instead explicitly show the x-intercept, you can write it in the form y = m(x - c). Then the x-intercept is c because when x = c, y = m(c - c) = m*0 = 0. The slope is still m. (This x-intercept is unique as long as m is nonzero.)

Have a blessed, wonderful day!