SOLUTION: this question is concerned with the solution of the linear system, y = mx + b y = nx + c 1) if the system has an infinite number of solutions, discuss the relationships among th

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Question 283338: this question is concerned with the solution of the linear system,
y = mx + b
y = nx + c
1) if the system has an infinite number of solutions, discuss the relationships among the four constants.
Found 2 solutions by Theo, richwmiller:
Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
This would happen is m = n and b = c.

This means that the lines are identical.

If the lines are identical then they intersect at all points.

If the equations are in same standard form, then this would happen is the equations are multiples of each other.

An example:

5x + 7y = 35
10x + 14y = 70

To convert these equations into slope intercept form, we solve for y.

The first equation becomes:

5x + 7y = 35 is the original equation.
Subtract 5x from both sides of this equation to get:
7y = -5x + 35
Divide both sides of this equation by 7 to get:
y = (-5/7)x + (35/7)

the second equation becomes:
10x + 14y = 70 is the original equation.
Subtract 10x from both sides of the equation to get:
14y = -10x + 70
Divide both sides of the equation by 14 to get:
y = (-10/14)x + (70/14)
Simplify to get:
y = (-5/7)x + (35/7)

The equations that were multiples of each other in standard form become the same as each other in slope-intercept form.

If you graph both these equations, you will get the same line.

If you get the same line, then there are an infinite number of solutions that will satisfy both equations.


Answer by richwmiller(17219) About Me  (Show Source):
You can put this solution on YOUR website!
Theo did a great job explaining the problem.
However he is a little off on his conclusion.
"The equations that were multiples of each other in standard form BECOME (my caps) the same as each other in slope-intercept form."
They CAN become the same .
y = (-10/14)x + (70/14)
CAN become
y = (-5/7)x + (35/7)
y = (-10/14)x + (70/14) DOES NOT HAVE TO change.
Also one might be in decimal form such as .5 and the other might be a fraction such as 1/2