Find a value for k so that the line through (35, k) and (42, 27) is parallel to the line with equation y = x. Application

The correct answer is: Value of k for the given line is 20.

Step-by-step solution:-
y = x
∴ y = x + 0.
Comparing the above equation with standard form of a line i.e. y = mx + c, we get- m = 1 ......................... (Equation i)
The given line is parallel to y = x
and we know that parallel lines have equal slopes.
∴ Slope of the given line = slope of line (y = x)
∴ Slope of the given line = m = 1 ..................... (From Equation i) .................................................................... (Equation ii)
The given line passes through the point (35,k) & (42,27).
Hence, x1 = 35, y1 = k, x2 = 42 & y2 = 27
(y-y1) = (y2-y1) × (x-x1)
(x2-x1)
∴ (y-k) = (27-k) × (x- 35)
42-35
∴ y - k = 27 - k ×(x - 35)
7
∴ 7 * (y - k) = (27 - k) × (x - 35) ............................................................................ (Multiplying both sides by 7)
∴ 7y - 7k = 27x - 945 - kx + 35k
∴ 7y = 27x - k x + 35k + 7k - 945
∴ 7y = (27 - k) x + 42k - 945
∴ y = (27 - k)/7 x + 6k - 135 ........................... (Dividing both sides by 7) ........................................... (Equation iii)
Comparing Equation iii with the standard form of a straight line i.e. y = mx + c, we get-
m = (27 - k) / 7
∴ 1 = (27 - k) / 7 ..................................... (From Equation ii)
∴ 7 = 27 - k ...................................... (Multiplying both sides by 7)
∴ k = 27 - 7
∴ k = 20.