SOLUTION: express each equation in slope-intercept form. then determin, without silving the system, whether the system of equations has exactly one solution, none, or an infinite number

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Question 145018: express each equation in slope-intercept form. then determin, without silving the system, whether the system of equations has exactly one solution, none, or an infinite number
4x+6y=26
-6x-9y=-39
Found 3 solutions by jim_thompson5910, solver91311, Alan3354:
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

4x%2B6y=26 Start with the first equation


6y=26-4x Subtract 4+x from both sides


6y=-4x%2B26 Rearrange the equation


y=%28-4x%2B26%29%2F%286%29 Divide both sides by 6


y=%28-4%2F6%29x%2B%2826%29%2F%286%29 Break up the fraction


y=%28-2%2F3%29x%2B13%2F3 Reduce



So the equation is now in slope-intercept form (y=mx%2Bb) where the slope is m=-2%2F3 and the y-intercept is b=13%2F3



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-6x-9y=-39 Move onto the second equation


-9y=-39%2B6x Add 6+x to both sides


-9y=%2B6x-39 Rearrange the equation


y=%28%2B6x-39%29%2F%28-9%29 Divide both sides by -9


y=%28%2B6%2F-9%29x%2B%28-39%29%2F%28-9%29 Break up the fraction


y=%28-2%2F3%29x%2B13%2F3 Reduce



So the equation is now in slope-intercept form (y=mx%2Bb) where the slope is m=-2%2F3 and the y-intercept is b=13%2F3



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Since the slope and y-intercept for both equations are the same, this means that there are an infinite number of solutions (since one equation lies on top of the other)

Answer by solver91311(24713) About Me  (Show Source):
You can put this solution on YOUR website!
put your equations in slope-intercept form by solving for y, that is, arrange them so that you have them in the form y=mx%2Bb. Make certain that you have reduced all fractions to the lowest terms.

Once you have done that you will have two equations:

y=m%5B1%5Dx%2Bb%5B1%5D and y=m%5B2%5Dx%2Bb%5B2%5D


If m%5B1%5D%3C%3Em%5B2%5D, then the lines intersect and there is exactly one element in the solution set for the system.

If m%5B1%5D=m%5B2%5D and b%5B1%5D%3C%3Eb%5B2%5D, then the lines are different but parallel so the solution set to the system is empty.

If m%5B1%5D=m%5B2%5D and b%5B1%5D=b%5B2%5D, then the lines are the same line and there are an infinite number of elements in the solution set for the system.

Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
4x+6y=26
-6x-9y=-39
4x+6y = 26
6y = -4x + 26
y = -(2/3)x + 13/3
m = -2/3
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-6x-9y = -39
2x + 3y = 13
3y = -2x + 13
y = -(2/3)x +13/3
This is the same equation with a scalar multiplier. The 2nd eqn is 1.5 times the first, so no explicit solution is possible.
There are an infinite number of combinations of x and y that satisfy the eqn, any point on the line that represents the eqn.