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Answer:
Closure property
Over addition
If a and b are two whole numbers and a + b = c, then c is also a whole number. 3 + 4 = 7 (whole number).
Over subtraction
If a and b are two whole numbers and a − b = c, then c is not always a whole number. Take a = 7 and b = 5, a − b = 7 − 5 = 2 and b − a = 5 − 7 = −2 (not a whole number).
Over multiplication
According to this property, if two integers a and b are multiplied then their resultant a × b is also an integer. Therefore, integers are closed under multiplication
Over division
The closure property of the division tells that the result of the division of two whole numbers is not always a whole number. Whole numbers are not closed under division i.e., a ÷ b is not always a whole number. From the property, we have, 14 ÷ 7 = 2 (whole number) but 7 ÷ 14 = ½ (not a whole number).
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Answer࿐
Given,
3x+2y = -4 ----------(1)
2x+5y = 1 -----------(2)
¶ Find Point of Intersection of these 2 lines
Do 2×(1) - 3×(2)
6x+4y = -8 ----------(3)
6x+15y = 3 ---------(4)
-----------------------
-11y = -11
=> y = 1
Substitute in (1)
3x+2(1) = -4
=> 3x = -4-2
=> 3x = -6
=> x = -2
•°• Point of Intersection of the lines 3x+2y+4= 0 & 2x+5y-1= 0 is (-2,1)
¶ By using the Slope-Intercept form, find the form of equation of line passing through the point (-2,1)
y = mx + c
substitute x = -2 & y = 1
=> 1 = -2m + c
=> c = 1 + 2m
•°• The Required Equations of straight line is of form :
y = mx + 1 + 2m -----------(5)
¶ The perpendicular distance (or simply distance) 'd' of a point P(x1,y1) from Ax+By+C = 0 is given by
Given,
(x1,y1) = (-2,1) & d = 2
=> (4m+2)² = 2(m² + 1)
=> 16m² + 16m + 4 = 2m² + 2
=> 14m² + 16m + 2 = 0
=> 7m² + 8m + 1 = 0
Factorise the equation
=> 7m² + 7m + m + 1 = 0
=> 7m(m+1) + 1(m+1) = 0
=> (m+1)(7m+1) = 0
=> m = -1 and m = -1/7
Now Substitute m = -1 in (5)
=> y = (-1)x+1+2(-1)
=> y = -x + 1 - 2
=> y = -1 - x
(or)
=> -x - y -1 = 0
=> x + y + 1 = 0 ------------(6)
Substitute m = -1/7 in (5)
=> y = (-1/7)x + 1 + 2(-1/7)
=> y = -x/7 + (7-2)/7
=> y = (-x+5)/7
=> 7y = -x+5
or
=> -x - 7y + 5 = 0
=> x + 7y - 5 = 0 ----------(7)
•°• The Required equation of straight lines are :
x + y + 1 = 0 & x + 7y - 5 = 0
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