the vaule of (2ax-b) (2a+b) is
Answers
Answer:
How do you find 'x' and 'y' if (2ax+3by=a+2b ; 3ax+2by=2a+b)?
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2ax + 3by = a+2b….(1)
3ax + 2by = 2a+b….(2)
Adding (1) and (2)
5ax + 5by = 3a + 3b….(3)
Subtracting (1) from (2)
ax - by = a-b …………..(4)
so 5ax - 5by = 5a - 5b….(5)
Subtracting (5) from (3)
10by = 8b -2a; simplifying
y = (4b -a)/5b……(6)
Adding (3) and (5)
10ax = 8a -2b; simplifying
x= (4a - b)/5a….(7)
Check
Is 2ax + 3by = a+2b….(1)?
Left hand side ..Substitute valueof x from eqn.…(7) and y from eqn.…(6)
2a* [(4a - b)/5a] + 3b*[(4b -a)/5b]
= [8a - 2b + 12b - 3a]/5 = [5a - 10b ]/5 = a + 2b i.e. Right hand side
Proves
x= (4a - b)/5a and y = (4b -a)/5b are correct.
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We can use 3 methods to solve these equations
Substitution method
Elimination method
Cross multiplication method
I am using elimination method
Let
2ax+3by=a+2b…………(1)
3ax+2by=2a+b…………(2)
On multiplying(1) by 3 and(2)by 2
6ax+9by=3a+6b……….(3)
6ax+4by=4a+2b………..(4)
(3)-(4)
(6ax-6ax)+(9by-4by)=(3a-4a)+(6b-2b)
5by=-a+4b
y=(4b-a)/5b putting in equation(1)
2ax+3b(4b-a)/5b=a+2b
2ax=a+2b-(12b^2-3ab)/5b
2ax=(5ab+10b^2–12b^2+3ab)/5b
2ax=(8ab-2b^2)/5b
x=(4a-b)/5a
Now verify
Putting the value of x and y in equation(1)
LHS
2a(4a-b)/5a+3b(4b-a)/5b
=(8a-2b+12b-3a)/5
=(5a+10b)/5
=a+2b =RHS
..
.
2ax+3by=a+2b ….(1)
3ax+2by=2a+b …..(2)
(2) - (1) gives
ax - by = a - b
Implies 5ax - 5by = 5a - 5b …..(3)
(1)+(2)=>5ax + 5by = 3a + 3b …..(4)
(3)+(4)=> 10ax = 8a - 2b = 2(4a - b)
So x = (1/5a)(4a-b)
(4) -(3) gives 10by= -2a + 8b = 2(4b - a)
So y= (1/5b)(4b - a)