No solutions, one solution, infinitely many solutions?

Let's be very direct. The equations

2x+6y=17​
x+3y=17​

are not equivalent; they represent parallel lines, because when you multiply the second equation by 2, you get

2x+6y=17​
2x+6y=34​

When you subtract the first from the second, you get 0 = 17, which tells you there is no solution. Or if you solve each equation for y, you get

y = -1/3 x + 17/6​
y = -1/3 x + 34/6​

which are parallel lines, having the same slope but different y-intercepts.

So having the right-hand sides be equal did not make the equations equivalent so there would be infinitely many solutions.

But look at these equations:

2x+6y=34​
x+3y=17​

Here we can tell that they represent the same line because if we multiply the second by 2 (so that the LHS becomes the same), the RHS's also become the same. That is, the ratio of the RHS is the same as the ratio of the LHS.

Now go back to your problem:

2x - 3y = a​
4x - 6y = b​

What must be true of a and b to make these equations represent the same line, so that the system is inconsistent?
 
frctl said:
2x - 3y = a ......(1)
4x - 6y = b ......(2)

(1) 3y = 2x - a
y = (2/3)x - (a/3)
(2) 6y = 4x - b
y = (4/6)x - (b/6)
y = (2/3)x - (b/6)

The coefficient of Y and X is the same on both above equations
This means they have the same slope but do not intercept.

When will these equations represent the same line?
...
When will they represent parallel lines?
...
When will they represent intersecting lines?
...
Click to expand...
YES these lines have the same slope. Well done.

For them to be the same line then they must have the same y-intercept, right? So what does that tell you about -a/3 and -b/6?

For them to be parallel then they cant have the same y-intercept, right? So what does that tell you about -a/3 and -b/6?

For them to be intersecting (one point in common) they can't have he same gradient BUT they do! So can they ever be intersecting?
 
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