Question:
Find the constants a and b so that all the 4 lines whose equation are given by
x + y = -1
-x + 3y = -11
ax + by = 4
2ax - by = 2
pass through the same point .
Answer:
Given 4 lines whose equations are:
x + y = - 1 ---> Equation 1
-x + 3y = - 11 ---> Equation 2
ax + by = 4 ---> Equation 3
2ax - by = 2 ---> Equation 4
Properties/Concept:
In order for the 4 equations to pass thru the same point, it means the 4 equations will have a common coordinate (x,y). This common coordinate is what we call Point of Intersection.
Hence we need to find the Point of Intersection (x,y) of the 4 lines.
Step 1: Solve for (x,y) using Equation 1 and 2
x + y = - 1 ---> Equation 1
x = - 1 - y
Plug in value of x above to Equation 2
-x + 3y = - 11 ---> Equation 2
- ( -1 - y) + 3y = - 11
+1 +y + 3y = -11
4y + 1 = -11
4y = -11 - 1
4y = -12
y = -3
now solve for x using either Equation 1 or 2, and plugin value of y = -3
x + y = - 1 ---> Equation 1
x = - 1 - y
x = -1 - ( - 3 )
x = -1 + 3
x = 2
Now we know the Point of Intersection (x,y) of the 4 lines.
Point of Intersection = (2, -3)
Step 2: Solve for a and b using the Point of Intersection (2, -3)
using Equation 3 and 4, plugin the values of x = 2, y = -3 to get the equation of the line in terms of variable a and b,
ax + by = 4 ---> Equation 3
a(2) + b(-3) = 4
2a - 3b = 4 ---> Equation 3a
2ax - by = 2 ---> Equation 4
2a(2) - b(-3) = 2
4a + 3b = 2 ---> Equation 4a
now solve for a and b using the Equation 3a and 4a
2a - 3b = 4 ---> Equation 3a
2a = 4 + 3b
a = (4+3b)/2
Plug in value of a above to Equation 4a
4a + 3b = 2 ---> Equation 4a
4 [(4+3b)/2] + 3b = 2
2(4+3b) + 3b = 2
8 + 6b + 3b = 2
8 + 9b = 2
9b = 2 - 8
9b = -6
b = -6/9
b = -2/3
now solve for a using either Equation 3a or 4a, and plugin value of b = -2/3
4a + 3b = 2 ---> Equation 4a
4a + 3(-2/3) = 2
4a + (-2) = 2
4a - 2 = 2
4a = 2 + 2
4a = 4
a = 4/4
a = 1
Hence the value of a = 1 and b = -2/3
Just to continue the solution:
Finally, we have values of a and b, hence our Equation 3 and Equation 4 can now be written as,
ax + by = 4 ---> Equation 3
(1)x + (-2/3)y = 4
x - 2/3y = 4
2ax - by = 2 ---> Equation 4
2(1)x - (-2/3)y = 2
2x + 2/3y = 2
Therefore the 4 equations whose common point is (2, -3) are,
x + y = - 1 ---> Equation 1
-x + 3y = - 11 ---> Equation 2
x - 2/3y = 4 ---> Equation 3
2x + 2/3y = 2 ---> Equation 4
and the attached picture shows the graph of the 4 equations with their point of intersection (2, -3)
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#No to copy paste solution
#No to plagiarism
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