How many solutions does the system of equations have?
3x = –12y + 15 and x + 4y = 5
To find the number of solutions, we need to determine if the system of equations is consistent, inconsistent, or dependent. We can do this by checking the slopes of the two linear equations.
The first equation can be rewritten as:
3x + 12y = 15
To find the slope of this equation, we need to express it in the form y = mx + b, where m represents the slope.
12y = -3x + 15
y = -3/12x + 15/12
y = -1/4x + 5/4
Therefore, the slope of the first equation is -1/4.
The second equation is already in the form y = mx + b:
x + 4y = 5
4y = -x + 5
y = -1/4x + 5/4
The slope of the second equation is also -1/4.
Since the slopes of the two equations are equal, the system of equations is dependent. This means that there are infinitely many solutions.