graphing linear equations

hunter121

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Sep 15, 2020
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I'm having trouble understanding what the question below is asking and how I would graph the equations.

The equations below are linear equations of a system where a, b, and c are positive integers.
ay + bx=c
ay - bx=c

which of the following describes the graph of a at least one such system of equations in the standard (x, y) coordinate plane.

I. 2 parallel lines
II. 2 intersecting lines
III. a single line

(the answer is 2 intersecting lines)
 
I'm having trouble understanding what the question below is asking and how I would graph the equations.

The equations below are linear equations of a system where a, b, and c are positive integers.
ay + bx=c
ay - bx=c

which of the following describes the graph of a at least one such system of equations in the standard (x, y) coordinate plane.

I. 2 parallel lines
II. 2 intersecting lines
III. a single line

(the answer is 2 intersecting lines)
Can you calculate slopes of the lines created by those equations?

Please show us what you have tried and exactly where you are stuck.

Please follow the rules of posting in this forum, as enunciated at:


Please share your work/thoughts about this problem.​
 
Another way of looking at it- without calculating slopes. If two equations give lines that are parallel they never intersect so there is NO (x, y) point that satisfies both equations. If the two equations give the same line then EVERY (x, y) point on that line satisfies both equations. If the two equations give intersecting lines, they intersect in one point so there is ONE (x, y) point that satisfies both equations.

Here the equations are ax+ by= c and ax- by= c. Adding the two equations eliminates y and we have 2ax= 2c, as long as a is not 0, x= c/a. The ax+ by= c+ by= c so by= 0 and, as long as b is not 0, y= 0. The two lines intersect in the single point (c/a, 0).

(If a= 0, the equations are by= c and -by= c. If b is not 0 there is no solution, so the lines are parallel, unless c is 0 in which case the line y= 0- the graphs are the same line. If a is not 0 but b is 0, the equations are both ax= c. x= c/a so the graphs are the same line. If a and b are both 0, the equations are 0= c. If c is not 0 that is never true so there is no graph. If c= 0 the graph is the entire xy-plane!)
 
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