## Positive and Negative Slopes

Simplificatation, evaluation, linear equations, linear graphs, linear inequalities, basic word problems, etc.
luv2laugh
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Joined: Mon Jun 28, 2010 7:26 pm
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### Positive and Negative Slopes

I am a beginner at algebra. I understand the formulas for finding the slopes and solving linear equations. I don't understand all the ins and outs of it, but I do have a good working knowledge of the concepts. What I need is some direction. I have been asked the following question.

Consider a system of linear equations in which the slope of one line is positive and the slope of the other line is negative. Will the system always have a solution? Explain how you know.

I read that horizontal lines do not have a slope and dividing by zero is undefined, but does that mean it is not a negative slope? I could benefit from some clarification on the question, as well.

Martingale
Posts: 333
Joined: Mon Mar 30, 2009 1:30 pm
Location: USA
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### Re: Positive and Negative Slopes

luv2laugh wrote:I am a beginner at algebra. I understand the formulas for finding the slopes and solving linear equations. I don't understand all the ins and outs of it, but I do have a good working knowledge of the concepts. What I need is some direction. I have been asked the following question.

Consider a system of linear equations in which the slope of one line is positive and the slope of the other line is negative. Will the system always have a solution? Explain how you know.

I read that horizontal lines do not have a slope and dividing by zero is undefined, but does that mean it is not a negative slope? I could benefit from some clarification on the question, as well.

Another way of asking: If one line has positive slope and the other has negative slope, will the lines always intersect?

luv2laugh
Posts: 2
Joined: Mon Jun 28, 2010 7:26 pm
Contact:

### Re: Positive and Negative Slopes

luv2laugh wrote:I am a beginner at algebra. I understand the formulas for finding the slopes and solving linear equations. I don't understand all the ins and outs of it, but I do have a good working knowledge of the concepts. What I need is some direction. I have been asked the following question.

Consider a system of linear equations in which the slope of one line is positive and the slope of the other line is negative. Will the system always have a solution? Explain how you know.

I read that horizontal lines do not have a slope and dividing by zero is undefined, but does that mean it is not a negative slope? I could benefit from some clarification on the question, as well.

Thank you for the reply. That does give me a better understanding of the question. Our material only gives us the basic proceedures and main points. I haven't been able to find anything that defines this question with details. Do you have any suggestions for me?

I would assume that having two opposing slopes would have an intersection somewhere, but I'm not sure that is the correct answer or why it would be if it is.

Martingale
Posts: 333
Joined: Mon Mar 30, 2009 1:30 pm
Location: USA
Contact:

### Re: Positive and Negative Slopes

luv2laugh wrote:Thank you for the reply. That does give me a better understanding of the question. Our material only gives us the basic proceedures and main points. I haven't been able to find anything that defines this question with details. Do you have any suggestions for me?

I would assume that having two opposing slopes would have an intersection somewhere, but I'm not sure that is the correct answer or why it would be if it is.

If the slopes are different then they will intersect.

since if you have two lines $y=a_1x+b_1$ and $y=a_2x+b_2$ with $a_1\neq a_2$

then if we set the two equations equal to each other we get

$a_1x+b_1=a_2x+b_2$

so

$a_1x-a_2x=b_2-b_1$

and

$(a_1-a_2)x=b_2-b_1$

since $a_1\neq a_2$ we have that $a_1-a_2\neq 0$

so we can divide by it and get

$x=\frac{b_2-b_1}{a_1-a_2}$

this is the location of the intersection of the lines.