Advertisements
Advertisements
Question
Solve each of the following systems of equations by the method of cross-multiplication :
`x(a - b + (ab)/(a - b)) = y(a + b - (ab)/(a + b))`
`x + y = 2a^2`
Advertisements
Solution
The solution of the systems of equation by the method of cross-multiplication:
Here we have the pair of simultaneous equation
`x(a - b + (ab)/(a - b)) = y(a + b - (ab)/(a + b)) = 0`
`x + y = 2a^2 = 0`
By cross multiplication method we get
`x/((-2a^2)xx-((a + b)- (ab)/(a + b)) - 0) = (-y)/((-2a^2)xx((a - b) + (ab)/(a - b)) = 0)`
`= 1/((a - b)+ (ab)/(a - b) - (-((a + b) - (ab)/(a + b))))`
`x/((-2a^2)xx-((a+ b)^2 + ab)/(a + b)) = (-y)/((-2a^2)xx(((a - b)^2 + ab)/((a - b)))`
`= 1/((((a - b)^2 + ab)/(a - b)) - (-(((a + b)^2 - ab)/(a + b))`
`x/((-2a^2)xx-((a^2 + b^2 + 2ab) - ab)/(a + b)) = (-y)/((-2a^2)xx((a^2 + b^2 - 2ab) + ab)/(a - b))`
`= 1/((((a^2 + b^2 - 2ab) + ab)/(a - b)) - (-(((a^2 + b^2 + 2ab) - ab)/(a + b))`
`x/(((2a^4 + 2a^2b^2 + 2a^3b))/(a + b))= y/((2a^4 + 2a^2b^2 - 2a^3b)/(a - b))`
`= 1/(((a^2 + b^2 -ab )(a + b) + (a^2 + b^2 + ab)(a - b))/((a - b)(a + b)))`
`x/((2a^4 + 2a^2b^2 + 2a^3b)/(a + b)) = y/((2a^4 + 2a^2b^2 - 2a^3b)/(a - b)) = 1/(((2a^3)/((a - b)(a + b)))`
Consider the following
`x/((2a^4 + 2a^2b^2 + 2a^3b)/(a + b)) = 1/(((2a^3)/((a - b)(a + b))))`
`x = ((a^2 + b^2 + ab)(a - b))/a`
`x = ((a^2 + b^2 + ab)(a - b))/a`
`x = (a^3 + ab^2 + a^2b - b^3 -ab^2 - a^2b)/a`
`x = (a^3 - b^3)/a`
And
`y/((2^4 + 2a^2b^2 -2a^3b)/(a - b)) =1 /((2a^3)/((a -b)(a + b)))`
`y/((a^2 + b^2 - ab)/(a - b)) = 1/(a/((a - b)(a + b)))`
`y(a/((a - b)(a + b))) = (a^2 + b^2 - ab)/(a - b)`
`y = ((a^2 + b^2 - ab)(a + b))/a`
`y = (a^3 + b^3)/a`
Hence we get the value of `x = (a^3 - b^3)/a and y = (a^3 + b^3)/a`
RELATED QUESTIONS
A train travels at a certain average speed for a distance of 54 km and then travels a distance of 63 km at an average speed of 6 km/h more than the first speed. If it takes 3 hours to complete the total journey, what is its first speed?
Solve the follownig system of equations by the method of cross-multiplication.
2x – 6y + 10 = 0
3x – 7y + 13 = 0
Solve the following system of equations by cross-multiplication method x + y = a – b; ax – by = a2 + b2
Solve the following system of equations by the method of cross-multiplication:
`\frac{x}{a} + \frac{y}{b} = a + b; \frac{x}{a^{2}} + \frac{y}{b^{2}} = 2`
Solve the following system of equations in x and y by cross-multiplication method
`(a – b) x + (a + b) y = a^2 – 2ab – b^2`
`(a + b) (x + y) = a^2 + b^2`
For which value of k will the following pair of linear equations have no solution?
3x + y = 1
(2k – 1)x + (k – 1)y = 2k + 1
Form the pair of linear equations in the following problems and find their solutions (if they exist) by any algebraic method:
Yash scored 40 marks in a test, getting 3 marks for each right answer and losing 1 mark for each wrong answer. Had 4 marks been awarded for each correct answer and 2 marks been deducted for each incorrect answer, then Yash would have scored 50 marks. How many questions were there in the test?
Form the pair of linear equations in the following problems and find their solutions (if they exist) by any algebraic method
The area of a rectangle gets reduced by 9 square units, if its length is reduced by 5 units and breadth is increased by 3 units. If we increase the length by 3 units and the breadth by 2 units, the area increases by 67 square units. Find the dimensions of the rectangle.
Solve the following system of equations:
0.5x + 0.7y = 0.74
0.3x + 0.5y = 0.5
Solve the following system of equations by the method of cross-multiplication:
`(x + y)/(xy) = 2`
`(x - y)/(xy) = 6`
Solve the following system of equations by the method of cross-multiplication:
ax + by = a2
bx + ay = b2
Solve the following system of equations by the method of cross-multiplication:
`(a - b)x + (a + b)y = 2a^2 - 2b^2`
(a + b)(x + y) = 4ab
Solve the following system of equations by the method of cross-multiplication:
`57/(x + y) + 6/(x - y) = 5`
`38/(x + y) + 21/(x - y) = 9`
Solve the system of equations by using the method of cross multiplication:
2x + y – 35 = 0, 3x + 4y – 65 = 0
Solve the system of equations by using the method of cross multiplication:
`(ax)/b - (by)/a - (a + b) = 0, ax - by - 2ab = 0`
Find the values of x and y in the following rectangle [see figure].
Solve the following pair of equations:
`4x + 6/y = 15, 6x - 8/y = 14, y ≠ 0`
In a competitive examination, one mark is awarded for each correct answer while `1/2` mark is deducted for every wrong answer. Jayanti answered 120 questions and got 90 marks. How many questions did she answer correctly?
Ankita travels 14 km to her home partly by rickshaw and partly by bus. She takes half an hour if she travels 2 km by rickshaw, and the remaining distance by bus. On the other hand, if she travels 4 km by rickshaw and the remaining distance by bus, she takes 9 minutes longer. Find the speed of the rickshaw and of the bus.
Anuj had some chocolates, and he divided them into two lots A and B. He sold the first lot at the rate of ₹ 2 for 3 chocolates and the second lot at the rate of ₹ 1 per chocolate, and got a total of ₹ 400. If he had sold the first lot at the rate of ₹ 1 per chocolate, and the second lot at the rate of ₹4 for 5 chocolates, his total collection would have been ₹460. Find the total number of chocolates he had.
