Solving two variable systems

[homeschool girl]

Suppose

is a constant, such that no pair of real numbers

satisfies this system of equations:

What is the value of

?

 

[MarkFL]

Consider two lines in the plane...what condition(s) do we require such that the lines never intersect?

 

[homeschool girl]

they must be parallel, or in other words, must have the same slope.

 

[MarkFL]

they must be parallel, or in other words, must have the same slope.

Yes, that is one condition. The other is that they cannot share any points in common, otherwise they will be the same line. Try arranging both lines in slope-intercept form (so we can require the slopes to be the same while making sure the intercept is different)...what do you get?

 

[homeschool girl]

y=(3/2)x+(7/4)

and

y=(K/8)x+(7/8)

 

[MarkFL]

I actually get:

[MATH]y=-\frac{3}{2}x+\frac{7}{4}[/MATH]
[MATH]y=-\frac{K}{8}x+\frac{7}{8}[/MATH]
Do you see where the negative sign comes from? Another way to approach this problem is to observe that given two linear equations of the form:

[MATH]Ax+By=C[/MATH]
[MATH]Ax+By=D[/MATH]
These lines will be parallel, and as long as \(C\ne D\) the lines will be distinct (not the same line).

 

[homeschool girl]

ahh, I see why it's negative. so K would be 12?

 

[MarkFL]

ahh, I see why it's negative. so K would be 12?

Yes.