*how many solutions does the system of equations have? 3x=-12y+15 and x+4y=5 *

The equations are,

3x = -12y + 15 ----------------- (1)

X + 4y = 5 ----------------------- (2)

Rearrange the 1st equation 3x = -12y + 15 as 3x + 12y = 15.

3x + 12y = 15 ----------------- (1)

X + 4y = 5 ----------------------- (2)

Now take out a factor of 3, giving,

3(x + 4y) = 15 ----------------- (1)

X + 4y = 5 ----------------------- (2)

Finally, divide by 3,

x + 4y = 5 ---------------------- (1)

X + 4y = 5 ----------------------- (2)

i.e. the 1st equation and the 2nd equation are identical.

That means that our two equations represent straight lines which are collinear.

The two lines lie on top of each other, which means that they intersect at an infinite number of points. i.e. any point on the 1st equation also satisfies the 2nd equation.

**So, your assumption is correct – there are in infinite number of solutions.**