The system of equations has how many solutions?
3x − 6y = 9
2y − x − 3 = 0
Number of solutions
First of all, notice that the slopes of these two lines are the same: both lines have a slope of 1/2. The lines are parallel. That leaves only two possibilities:
Case I: the lines are parallel and separate; they never intersect, so there are zero solutions.
Case II: the lines are not just parallel but in the exact same place; they overlap completely, from negative infinity to positive infinity, so they have infinitely many solutions.
Once we realize that slopes are equal, how do we distinguish these two cases? We have three ways.
(1) We can simply solve algebraically for the solution, using the techniques for simultaneous equations. If we get a tautology (i.e. a statement that is always true, such as 3 = 3), then the intersections are true everywhere, and it's Case II. If we get an impossibility (i.e. a statement that is never true, such as 4 = 9) , then the intersections are true nowhere, and it's Case I.
(2) We can put both lines in slope-intercept form. We know the slopes are equal. If the y-intercepts are different, then we have two different parallel lines, and we are in Case I. If the y-intercepts are equal, then both lines are in the same place, and we are in Case II.
(3) We could graph the two lines, and see directly.
First, let's solve these two algebraically.
Equation A: 3x − 6y = 9
Equation B: 2y − x − 3 = 0
Solve equation B for x:
x = 2y – 3
Now, substitute this into equation A:
3(2y – 3) – 6y = 9
6y – 9 – 6y = 9
–9 = 9
0 = 18
These final two statements are impossibilities, so we are in Case I.
A different method of solution.
Let's put them in slope-intercept form (i.e. y = mx + b form).
The slopes of course are equal, but the y-intercepts are not equal. The lines are parallel in different places, so again, we are in Case I.
Finally, let's look at a graph of these functions:
Clearly, the lines are parallel and in different places. They never intersect, so there are no solutions to the system of equations. Again, Case I. Zero solutions.
Answer = A
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