For what value of the constant (c) does the system of equations below have no solutions (x,y)?
3x - 5y = -2.3
6x - cy + 9.3
How would I start to figure this out? Thank you!
its 10
To determine the value of the constant (c) for which the system of equations has no solutions (x, y), we can use the concept of slopes.
The given system of equations can be rewritten in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept.
The first equation: 3x - 5y = -2.3
Rearranging the terms gives: -5y = -3x - 2.3
Dividing by -5: y = (3/5)x + 0.46
The second equation: 6x - cy + 9.3
Rearranging the terms gives: cy = 6x + 9.3
Dividing by c: y = (6/c)x + (9.3/c)
For the system to have no solutions, the lines represented by the equations must be parallel, meaning they should have the same slope.
Comparing the slopes:
The first equation has a slope of 3/5.
The second equation has a slope of 6/c.
For the lines to be parallel, the slopes should be equal. Therefore, we can set the slopes equal to each other:
3/5 = 6/c
To solve for c, we can cross-multiply:
3c = 5 * 6
3c = 30
Dividing by 3:
c = 10
So, the value of the constant (c) for the system of equations to have no solutions (x, y) is 10.
To determine the value of the constant (c) for which the system of equations has no solutions, we need to consider the coefficients of the variables (x and y) in the two equations and check if they are proportional or not.
Let's start by analyzing the coefficients of y in both equations. In the first equation, the coefficient of y is -5, and in the second equation, the coefficient of y is -c.
For the system of equations to have no solution, the coefficients of y in both equations must be proportional with different constant factors. In other words, -5 should not be equal to -c or -5 ≠ -c.
To find the value of c for which -5 ≠ -c, we can solve the equation -5 ≠ -c for c.
Adding c to both sides of the equation, we get: -5 + c ≠ 0.
To isolate c, we subtract 5 from both sides of the equation: c - 5 ≠ 0.
Thus, we conclude that the system of equations has no solution for any value of c ≠ 5. This means that for the constant c to make the system unsolvable, c must not equal 5.
So, the value of the constant (c) for which the system of equations has no solutions is c ≠ 5.