Question

Linear equations, can someone please help explain this?

Answer 1

What is an equation that can be written with 10x+5y=20 to form a pair of linear equations that has exactly one solution.

Answer 2

well...we know it can't have the same slope....because if it had the same slope and different y intercepts, then it would be parallel with no solutions...and if it had the same slope and the same y intercept, then it would be the same line with infinite solutions.
So, lets find the slope of your equation. In y = mx + b form, the slope will be found in the m position.
10x + 5y = 20 --- subtract 10x from both sides
5y = -10x + 20 -- divide both sides by 5
y = -2x + 4
so our slope is -2....you cannot have -2 as your slope.
so here is one...
y = 5x + 6 -- turn to standard form of Ax + By = C
-5x + y = 6 -- multiply by -1
5x - y = -6 <=== here is another equation that will result in 1 solution

Answer 3

did I lose you ?

Answer 4

Lol no and thanks! what about one with an endless amount of solutions

Answer 5

endless amount of solutions....infinite solutions.....that would mean it is the same line.
Example :
10x + 5y = 20
2x + y = 4

if you were to divide the first equation by 2, you would get : 2x + y = 4...making both lines be the same...producing infinite solutions

Answer 6

oops...divide the first equation by 5

Answer 7

You are awesome my teacher makes this seems complicated! Last one, what about if there are no solutions? and how many possible answers are there to all of these.

Answer 8

no solutions....that would mean they are parallel lines. Parallel lines will contain the same slope, but different y intercepts.
10x + 5y = 20
we have already established that the slope in the equation is -2...if we want " no solutions " we want a parallel line...so we want a slope of -2...and a different y intercept.
In y = mx + b, the slope is in the m position and the y intercept is in the b position.
So we want -2 as our slope and any number besides 20 as our y intercept.
like this :
y = -2x + 10 -- turn to standard form
2x + y = 10 --- this will produce a parallel line with no solutions

Answer 9

oops...kinda messed up
10x + 5y = 20
5y = -10x + 20
y = -2x + 4
we would want slope of -2 and any number besides 4 as our y intercept

Answer 10

Just learn this...it will help :
no solution -- parallel lines..same slope, different y intercepts
one solution -- different slopes,does not matter if y intercepts are the same or not
infinite solutions -- same line....same slopes, same y intercepts