Determine the number of solutions for the system without solving.
3x+12y=-27
4x+2y=34
Transform each equation to the form
y=mx+b
where m=slope and b=y-intercept.
If the slopes (m) are distinct (different), then there is one solution.
If the slopes are identical, then check the y-intercepts. If the y-intercepts are different, there is no solution (lines are parallel and distinct).
If the slopes are identical and the y-intercepts are the same, then there are infinite solutions (lines are coincident).
In this case, there is one solution.
How many solutions are there to the following system of equations?
3x-9y=0
-x+3y=-3
To determine the number of solutions for a system of equations without actually solving them, we can consider the slopes of the lines represented by the equations.
First, let's rewrite the equations in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept.
Equation 1: 3x + 12y = -27
Rearranging, we have: 12y = -3x - 27
Dividing by 12, we obtain: y = (-1/4)x - 9/4
This equation has a slope of -1/4.
Equation 2: 4x + 2y = 34
Rearranging, we get: 2y = -4x + 34
Dividing by 2, we have: y = -2x + 17
This equation has a slope of -2.
Now, if the slopes of the two lines are different, then the system has a unique solution. If the slopes are the same, the system has either infinitely many solutions (if the lines are the same) or no solution (if the lines are parallel).
In this case, the slopes are different (-1/4 and -2), so the system has a unique solution.
Therefore, the number of solutions for the system is 1.