How many solutions does this system of equations have?
y=3x+6 and y=-13x-4
To find the number of solutions, we need to determine whether the two lines represented by the equations intersect, are parallel, or coincide (are the same line).
The system of equations is:
1) y = 3x + 6
2) y = -13x - 4
The slopes of the lines are the coefficients of x, which are 3 in equation 1) and -13 in equation 2). Since the slopes are different, the lines are not parallel.
To determine if the lines intersect at one point, we can set the two equations equal to each other:
3x + 6 = -13x - 4
Simplifying:
16x = -10
x = -10/16
Simplifying the fraction by canceling the common factor of 2:
x = -5/8
Substituting x = -5/8 into either of the original equations, we can solve for y:
Using equation 1) y = 3x + 6:
y = 3(-5/8) + 6
y = -15/8 + 48/8
y = 33/8
Therefore, the system of equations has one solution at (-5/8, 33/8).