SOLUTION: if the graph of two linear equations in a system have different slopes will the system always have exactly one solution?

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Question 73044: if the graph of two linear equations in a system have different slopes will the system always have exactly one solution?
Found 2 solutions by jim_thompson5910, bucky:
Answer by jim_thompson5910(35256) (Show Source):
You can put this solution on YOUR website!
Yes it will since a line extends infinitely in each direction, it is bound to intersect with another line of a different slope.

Answer by bucky(2189) (Show Source):
You can put this solution on YOUR website!
This is true if you don't place any limits on the domain of x or the range of y.
.
There are three possibilities that can occur with linear equations. They are:
.
(1) The first possibility is that the graphs are parallel and different lines. This implies
that they have the same slopes but are separated by some amount of vertical distance.
Example:
.
y = 3x + 7
y = 3x - 2
.
These two equations have parallel graphs because they both have a slope of +3,
but they
are separated by 9 vertical units. (One crosses the y-axis at +7 and the other crosses
the y-axis at -2). In this possibility, because the graphs are parallel, there is no common
solution.
.
(2) The second possibility is that the graphs lie on top of each other so that every solution
of one of the equations is a solution of the other equation also.
.
Example:
.
y = 5x = 14
2y = 10x + 28
y = (1/2)*(10x + 28)
.
With a little manipulation you will find that these three are all the same equation and
therefore their graphs are on top of each other. So a solution for one of them is a solution
for all of them.
.
(3) The third possibility is that the graphs have different slopes. Two such lines can
intersect at only one point, and that point is the common solution. However, if you put limits
on the values of x and y in addition to the equation, then there may not be a common solution.
.
Example:
.
Given the equations:
.
y = 2x + 4 and
y = x + 5
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If you work this out you will find that the common solution is x = 1 and y = 6.
(Note: these
two equations have different slopes. One has a slope of 2 and the other has a slope of +1.)
.
But if the problem had said:
.
Find the common solution of:
.
y = 2x + 4 and
y = x + 5
.
If the x value must be negative and the y value cannot be less than 7. In that case there
is no common solution because we know that in the common solution the x value is positive 1
and the y value is positive 6. Both of these limits (the x value limit and the y value limit)
cannot be satisfied by the common solution values for x and y.
.
Hope this last part doesn't confuse you. For beginners it is probably enough to say that
with different slopes there can and will be only one common solution to two linear
equations.