The standard form of the equation of the line through the pair of points (10,5) and (-2,10) write the equation using only integer coefficients is 5x + 12y -110 = 0.
Given: The pair of points (10, 5) and (-2, 10). To write the equation of a line in standard form.
We know that the standard equation of a line is given by the formula:
Ax + By - C = 0, where A and B are coefficients of x and y respectively and C is a constant.
The point-slope from a line passing from a given (x₁, y₁) point is given as:
(y - y₁) = m(x - x₁), for a point (x₁, y₁), and m is the slope.
When a given line passes from two points (x₁, y₁) and (x₂, y₂), the slope m is given by the formula:
m = (y₂ - y₁) / (x₂ - x₁)
Therefore slope of the line passing from the points (10, 5) and (-2, 10) is given as: [x₁ = 10, y₁ = 5, x₂ =-2, y₂ =10]
m = (10 - 5) / (-2 -10)
= 5 / -12
m = -5 / 12
Now the equation of the line in point-slope form is given as (y - y₁) = m(x - x₁)
Putting values of m = -5 / 12 and (x₁, y₁) = (10, 5) in the equation above, we get:
y - 5 = -5 / 12 (x - 10)
Multiplying 12 on both sides:
12 × (y - 5) = 12 × -5 / 12(x - 10)
12y - 60 = -5(x - 10)
12y - 60 = -5x + 50
Adding 5x on both sides:
12y - 60 + 5x = -5x + 50 + 5x
5x + 12y - 60 = 50
Adding -50 on both sides:
5x + 12y - 60 + (-50) = 50 + (-50)
5x + 12y -110 = 0
Therefore the required equation is 5x + 12y -110 = 0.
Hence the standard form of the equation of the line through the pair of points (10,5) and (-2,10) write the equation using only integer coefficients is 5x + 12y -110 = 0.
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