Determine the number of solutions of the linear system:
14x + 7y = 315
16x - 2y = 610

Question

anonymous · Answer

elimination method or substitution

anonymous · Answer

Since the system is linear and the lines do not intersect, there can only be one solution.

anonymous · Answer

if they do not intersect then they have no solution...

anonymous · Answer

if you multiply the first by 2 and the second by 7 you get

$$28x+14y=630$$
$$112x-14y=4270$$

adding these you get

$$140x=4900$$

$$x=35$$

anonymous · Answer

find y by plugging in x=35 to one of the above equations.. This makes total sense because both are linear equations, so they're lines. the lines will meet only at one point because if you continuously draw a two lines.|dw:1342807258681:dw|

anonymous · Answer

they will only hit once, never hit, or be the same line

anonymous · Answer

does this make sense?

anonymous · Answer

well yes.. but I'm a little confused... is it possible to determine  if they only hit once, never hit, or are the same line?

anonymous · Answer

yes if they're the same line, one line is a linear combination of the other. What this means is

$$c_1x+c_2y=k$$

anonymous · Answer

so say you have a two lines

$$2x+4y=10$$
$$x+2y=5$$

these are the same line because if you multiply the bottom by 2 you get

$$2x+4y=10$$
$$2x+4y=10$$

anonymous · Answer

in this case if you have two exact lines, they hit the same exact points infinitely which means there is tons of solutions

anonymous · Answer

No solution happens when you have two lines that when you add them using elimination you cancel one side out and you get a false statement. An example of this is two lines with the same slope having different y intercepts

anonymous · Answer

$$2y+2x=400$$
$$2y+2x=500$$

if you multiply the second by -1 and add them you get

$$0=-100$$
this is false because 0 does not equal -100. In this case there is no solution the picture of this would be
|dw:1342808136345:dw|

anonymous · Answer

lastly like your question if when you eliminate you're able to solve for one variable, there is one solution to the system

anonymous · Answer

$$2x+4y=10$$
$$x+5y=25$$

multiply the last by -2 and add both equations

$$2x+4y=10$$
$$-2x-10y=-50$$

$$-6y=-40$$

$$y=40/6$$

anonymous · Answer

now use this in one of thebeginning equations to find x i'll pick the first one

$$2x+4(40/6)=10$$
$$2x+(160/6)=(60/6)$$
$$2x=(-100/6)$$
$$x=(-100/12)$$

anonymous · Answer

so the point at which they intersect is 

$$(-100/12,40/6)$$

anonymous · Answer

Okay great answer!! Thank you :)

anonymous · Answer

no proble

anonymous · Answer

could you also answer my other graphing question ?? haha.. :)