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
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).
To find the value of the constant c that makes the system of equations have no solution, we need to determine when the two equations are parallel. This happens when the slopes of the two lines are equal.
The slope of the first equation can be found by rearranging the equation in the form y = mx + b:
3x - 5y = -2.3
-5y = -3x - 2.3
y = (3/5)x + 2.3/5
The slope of the second equation can be determined by solving it for y:
6x = cy + 9.3
cy = 6x - 9.3
y = (6/c)x - 9.3/c
For the two lines to be parallel, their slopes must be equal:
(3/5) = 6/c
Now, we can solve the equation (3/5) = 6/c for c:
Multiply both sides by 5 to eliminate the fraction:
(3/5) * 5 = (6/c) * 5
3 = 30/c
Divide both sides by 3:
3/3 = (30/c) /3
1 = 10/c
Multiply both sides by c:
c = 10
Therefore, the value of the constant c that makes the system of equations have no solution is c = 10.
To find the value of the constant c that makes the system of equations have no solution, we need to analyze the coefficients and constants of the equations. Here's how you can solve it:
1. Rewrite both equations in slope-intercept form, which is y = mx + b.
First equation:
3x - 5y = -2.3
-5y = -3x - 2.3
y = (3/5)x + 0.46
Second equation:
6x = cy + 9.3
cy = 6x - 9.3
y = (6/c)x - (9.3/c)
2. Compare the slopes (coefficients of x) of the two equations.
The slope of the first equation is 3/5.
The slope of the second equation is 6/c.
3. If two lines have different slopes, they will intersect at one point, forming a unique solution. However, if their slopes are equal, the lines are either the same or parallel.
In this case, for the system of equations to have no solution, we need the slopes to be equal, but not the same. Therefore, we need:
3/5 = 6/c
4. Solve for c.
Multiply both sides of the equation by 5c:
3c = 30
Divide both sides of the equation by 3:
c = 10
Therefore, the constant c must be equal to 10 to make the system of equations have no solution.