# Math topic: graph a system of equations

Discussion in 'Middle School / Junior High' started by NoviceTutor, Mar 19, 2015.

1. ### NoviceTutorCompanion

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Mar 19, 2015

A student is asked to graph two lines from a given solution of (-2,1) on a grid paper. Can it be done? Typically it takes two points to create a line. A general linear equation is y - (b) = m * [x - (x)]. I am really grateful if some mathematicians out there could help with this question.

2. 3. ### mathmagicEnthusiast

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Mar 19, 2015

There are an infinite number of lines that pass through a single point, such as (-2, 1). In fact, if you were to give a second point, it would then limit it to just one line.

Could the question have asked the student to find any line that has that as a given solution? Alternatively, could they be wanting the student to find two lines through the point that are perpendicular to each other? That would be slightly less trivial.

(Or maybe it wasn't even a question, and I'm making a mountain out of a mole hill. Oh well, I love math. )

4. ### NoviceTutorCompanion

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Mar 19, 2015

Thanks Mathmagic for the feedback. I double checked the direction and it says, "graph two linear equations that intersects at the point (-2,1)." What is so weird is that only one ordered pair is given. Is there a way to graph 2 lines from a single ordered pair? I am used to seeing two given linear equations in the form of y = mx + b to arrive at a solution of (x,y), but from (x,y) to the lines is very rare to me.

For example, find a solution to the following system of equations: X + y = 7; x - y = -1. When graphed, the two lines intersect at the ordered pair (3, 4). However, if the process were to be reversed from (3, 4) to the two lines, I would find it rather challenging. This is where I am stuck.

5. ### mathmagicEnthusiast

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Mar 19, 2015

Yes. You can find all possible lines through first plugging the values of each coordinate into y=mx+b:

1 = -2m + b
b = 2m+1

Now you can pretty much decide whatever slope you want your line to have, which will give you the y-intercept (b), which then allows you to write the equation for one of the lines by plugging in your chosen value of m (slope) and the value of the y-intercept that popped out.

For example:

m = 2
b = 2(2) + 1 = 5
so one equation for a line is y = 2x + 5

m = 1
b = 2(1) + 1 = 3
so another equation for a line through that point is y = x + 3

And right there, you have two equations that will intersect at that point, because they both are guaranteed to go through that point.

6. ### MikeTeachesMathDevotee

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Mar 19, 2015

y = 3x + 7

y = -0.5x

Two lines that intersect at (-2, 1). Infinitely more are possible.

y = mx + b

1 = m(-2) + b

Pick and choose any numbers. Say m = 100.

1 = (100)(-2) + b

1 = -200 + b

b = 201

y = 100x + 201

See?

7. ### NoviceTutorCompanion

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Mar 19, 2015

Mathmagic, you are absolutely the best. I finally see the two lines as outlined by your explanation. Thanks.

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