Answer: Two lines are coincident when they are essentially the same line, meaning they have the same slope and y-intercept. To find the value of k that makes the two given equations coincident, we need to check if the slopes and y-intercepts are the same.
Given equations:
2x + 3y = 5
5x + 15 - 2y = k
Let's rewrite equation 2 in the standard form (Ax + By = C):
2y = 5x + 15 - k
Divide both sides by 2:
y = (5/2)x + (15 - k)/2
Now we can compare the equations to find the slope and y-intercept.
Slope (m1) of equation 1: Coefficient of x = 2
Slope (m2) of equation 2: Coefficient of x = 5/2
The slopes are not equal, so the lines are not coincident.
However, let's check the y-intercepts.
Y-intercept (b1) of equation 1: Coefficient of y = 3
Y-intercept (b2) of equation 2: Constant term = (15 - k)/2
For the lines to be coincident, the y-intercepts must be equal.
So, we equate the y-intercepts and solve for k:
3 = (15 - k)/2
Multiply both sides by 2:
6 = 15 - k
Now, isolate k:
k = 15 - 6
k = 9
Therefore, the value of k that makes the two equations represent coincident lines is k = 9.