Solve the linear system and determine points of intersection

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frctl

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Solve the linear system, if possible, and use the result to determine whether the lines represented by the equations in the system have zero, one, or infinitely many points of intersection. If there is a single point of intersection, give its coordinates, and if there are infinitely many, find parametric equations for them.

3x - 2y = 4
6x - 4y = 9
 
You mention "Linear Algebra". Are you taking a Linear Algebra course? Typically, one deals with "simple" systems of equations like this in a basic Algebra or Intermediate Algebra. You can solve (or "try to solve" if there is no one solution) by reducing the two equations in two unknowns to one equation in one unknown. For example you could multiply each equation by numbers so one of the unknowns has the same coefficient, then subtract one from the other. Here I see that y, in the first equation, has coefficient "-2" and in the second equation coefficient -4y. So start by multiplying the first equation by 2, to get 6x- 4y= 8. What do you get if you subtract that from the second equation? For what values of x is that true?
 
frctl said:
Solve the linear system, if possible, and use the result to determine whether the lines represented by the equations in the system have zero, one, or infinitely many points of intersection. If there is a single point of intersection, give its coordinates, and if there are infinitely many, find parametric equations for them.

3x - 2y = 4
6x - 4y = 9
Click to expand...
You are given two equations and two unknowns (x & y).

Solve for those.

Those are the co-ordinates of the point of intersection.

What do you get?

Please share your work with us - even if you know it is incorrect.
 
frctl said:
Solve the linear system, if possible, and use the result to determine whether the lines represented by the equations in the system have zero, one, or infinitely many points of intersection. If there is a single point of intersection, give its coordinates, and if there are infinitely many, find parametric equations for them.
\(\displaystyle 3x - 2y = 4\\6x - 4y = 9\)
Click to expand...
Can you show that \(\displaystyle \frac{3}{2}\) is the slope of both lines?
If you do, what does that tell you?
 
I have tried to reduce:
Multiply first eqn by 2
6x - 4y = 8
6x - 4y = 9
Subtract both eqns
8y = -1 (is this correct?)

I don't see how you obtained 3/2 for the slope.
Thank you for taking the time to explain to me.
 
frctl said:
I have tried to reduce:
Multiply first eqn by 2
6x - 4y = 8
6x - 4y = 9
Subtract both eqns
8y = -1 (is this correct?)

I don't see how you obtained 3/2 for the slope.
Thank you for taking the time to explain to me.
Click to expand...
Do you know the equation of a straight line:

y = m * x + b

What is the slope here?
 
frctl said:
I have tried to reduce:
Multiply first eqn by 2
\(\displaystyle 6x - 4y = 8\\6x - 4y = 9\)
Subtract both eqns
8y = -1 (is this correct?)NO!
Click to expand...
When we subtract we get \(\displaystyle \Large{0=-1}\)
What does that tell us?
 
Is the slope rise over run, therefore Δy / Δx..

That there is no x and no y coordinates and therefore no points of intersection.
 
If two lines have the same slope, that means they're parallel to each other. To solve this system of equations is equivalent to finding any point(s) where the lines intercept each other. So, your task is to synthesize all of this information. What do you know about parallel lines? Specifically what do you know about the intersection of parallel lines?
 
ksdhart2 said:
If two lines have the same slope, that means they're parallel to each other.
Click to expand...

If two lines (in the plane) have the same slope, that means either they're parallel to each other (they meet nowhere),
or they are the same line (they meet everywhere).

The phrase "two lines" does not necessarily mean "two distinct lines."


Parallel (geometry) - Wikipedia

en.wikipedia.org en.wikipedia.org
 
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lookagain said:
The phrase "two lines" does not necessarily mean "two distinct lines."
Click to expand...

Right, yes. You're technically correct (as you are most the time). However, human beings are not robots and are capable of extrapolating from incomplete information. In this specific case, the word "distinct" was unnecessary, because anybody who's been following the thread (and I sure hope the student has...) would know, from context, that the two lines in question are not the same line.
 
lookagain said:
The phrase "two lines" does not necessarily mean "two distinct lines."
Click to expand...
That is a false statement. If each of \(\displaystyle \ell_1~\&~\ell_2\) is a line and \(\displaystyle \ell_1=\ell_2\) then there is only one line.
In English two means distinct
 
pka said:
In English two means distinct
Click to expand...
This can be a little tricky!

If I talk about adding two numbers, does that imply they can't be the same number?

Or would you consider that a different issue?

If it's true that "two lines" must be distinct, then to talk about "two lines being coincident" would seen to be an oxymoron (or worse); but we do find that being said, for example here:

In three-dimensional Euclidean geometry, if two lines are not in the same plane they are called skew lines and have no point of intersection. If they are in the same plane there are three possibilities: if they coincide (are not distinct lines) they have an infinitude of points in common (namely all of the points on either of them); if they are distinct but have the same slope they are said to be parallel and have no points in common; otherwise they have a single point of intersection.​
 
In Steve Krantz's MATHEMATICAL Apocrypha on page 199 about a disagreement between E.H. Moore & Solomon Lefschetz concerning this very point.
EH Moore was RL Moore's thesis adviser and one of RL's students was mine. So it was drilled in to me over six years. You might call it tradition.
 
frctl said:
… That there is no x and no y coordinates and therefore no points of intersection.
Click to expand...
Is that your answer to pka's question in post #7?

? When you don't quote the specific post you're replying to, it's not always clear who you're talking to. (In your threads, you're skipping a number of questions that tutors have asked you.)
 
frctl said:
… I don't see how you obtained 3/2 for the slope …
Click to expand...
Working with 'slope' is unfamiliar because you've decided to study linear algebra without a background in beginning-algebra. You're missing also a second prerequisite, as you haven't yet finished learning pre-algebra topics (eg: arithmetic with ratios). What you're trying to do is self-study three full math courses at the same time. You're going to be busy for awhile!

In beginning algebra, we learn several ways to find slopes. When you're given a linear equation, an easy way is: Solve the equation for the dependent variable (in this exercise, y is the dependent variable). The equations were given in Standard form:

A∙x + B∙y = C

Subtract A∙x from each side:

B∙y = -A∙x + C

Divide each side by B:

y = (-A/B)∙x + C/B

The equation has been solved for y. We call this form 'Slope-Intercept', and, as Subhotosh showed in post #6, the symbols m (for the slope -A/B) and b (for C/B) are used:

y = mx + b

The line crosses the y-axis at the point (0,b). We call (0,b) the y-intercept. The form y=mx+b is called the Slope-Intercept form because it shows us both the line's slope (m) and its y-intercept (0,b) -- simply by looking at them.

frctl said:
Solve the linear system …
3x - 2y = 4
6x - 4y = 9
Click to expand...
Put each equation into Slope-Intercept form (i.e., solve for y). We can use either y=(-A/B)x + (C/B) or we can solve for y. I'll do both ways.

1st equation: 3x - 2y = 4

-A = -3
B = -2
C = 4
m = -A/B = -3/(-2)
b = B/C = 4/(-2)

y = (3/2)x - 2

2nd equation: 6x - 4y = 9

Subtract 6x from each side

-4y = -6x + 9

Divide each side by -4 (that's the same as multiplying each side by the reciprocal: -1/4)

y = (3/2)x - 9/4

The Slope-Intercept form shows us that both lines have the same slope (they are parallel) but cross the y-axis at different points (they are distinct lines).

So, you were correct (in post #8); the given lines do not intersect. Yet, your answer could be better written. Instead of saying "there is no x and no y coordinates", you need to be complete: "There are no (x,y) coordinates that solve this system".

You could also write "Inconsistent system" or "No solution".

?
 
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