How many solutions does this system of equations have 3x+2y=6, - 4x+5y=15

One

Step-by-step explanation:

A system would have no solution if the lines are parallel, which occur when they have the same slope (m value in y-intercept form y=mx+b)

A system would have an infinite number of solutions when the equations are equivalent. An equivalent equation is made by either rearranging or proportionately changing an equation. For example, if one equation is double the other, they are equivalent.

A system would have one solution in any other case.

3x+2y=6, - 4x+5y=15

Rearrange to y-intercept form:

3x+2y=6

2y = - 3x + 6

y = (-3/2) x + 6/2

y = (-3/2) x + 3

-4x+5y=15

5y = 4x + 15

y = (4/5) x + 15/5

y = (4/5) x + 3

There is one solution.

The slopes are not the same:

4/5 ≠ - 3/2

The equations are not equivalent. If they were, this would be true:

(-3/2) x + 3 = (4/5) x + 3

It is not true:

(-3/2) x + 3 = (4/5) x + 3

(-3/2) x = (4/5) x

(-15/10) x = (12/10) x

LS≠RS