direct variation, The amount (a) that a family gives to charity varies directly as its income (i). Last year, the family earned 32, 000 and gave 2560 to charity. How much will they give if they make 42,000 this year?
Answers
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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