How do you solve the following system of equations #x - 2y = 5# and #2x - 4y = 10#?
They are really the same equation with an infinite number of solutions
We could use substitution or the addition/subtraction method.
Let's use substitution:
We'll solve the first equation for
Substitute in the other equation, to get:
so we want
The way substitution works is to suppose we have an
When we get
Every solution to the first equation already is a solution to the second. There is no additional requirement.
If you think about the lines we'd get if we graphed these 2 equations, you'll see that they are the same line. Every point on one line is a point of the other line.
OK there's only one line so it might be better to say: every solution to one equation is a solution to the other.
There are a couple of ways to write the solution:
we can say "the system is dependent"
of, course that doesn't really say what the solutions are.
Every solution to the first equation,
Often, we like to do things by first choosing
We can now write the solutions:
If you write both equations in slope-intercept form:
(That's one reason we like slope-intercept form.)