Question

for what value of the constant c does the system of equations below have no solution (x,y) ?

3x-5y=-2.3
6x=cy+9.3

Answer 1

To determine when there is no solution to the system of equations, we can look at the slope of the lines represented by each equation. If the slopes are equal, then the lines are parallel and do not intersect, meaning there is no solution.

In the given system of equations:

Equation 1: 3x - 5y = -2.3
Equation 2: 6x = cy + 9.3

To find the slope of each equation, we can rewrite them in slope-intercept form by solving for y:

Equation 1:
3x - 5y = -2.3
-5y = -3x - 2.3
y = (3/5)x + (2.3/5)

Equation 2:
6x = cy + 9.3
cy = 6x - 9.3
y = (6/c)x - (9.3/c)

Now we can observe the slopes:
The slope of Equation 1 is 3/5, and the slope of Equation 2 is 6/c.

For the system of equations to have no solution, the slopes must be equal, meaning:
3/5 = 6/c

To solve for c, we can cross-multiply and solve for c:
3c = 6 * 5
3c = 30
c = 30/3
c = 10

Therefore, for c = 10, the system of equations has no solution (x, y).