Question

Q1.solve by cross multiplication method. x+y=a+b,ax-by=a^2-b^2​

Answer 1

Step-by-step explanation:

Given:

$\begin{gathered}x + y = a + b \\ ax - by = {a}^{2} - {b}^{2} \\ \end{gathered}$

To find: Solve linear equations using cross multiplication method.

Solution:

If

$a_1x+b_1y+c_1=0$

and

$a_2x+b_2y+c_2=0$

are linear equations in two variables, then

using cross multiplication method

$\begin{gathered}\frac{x}{b_1c_2-b_2c_1}=\frac{y}{a_2c_1-a_1c_2}=\frac{1}{a_1b_2-a_2b_1}\\\end{gathered}$

or

$\begin{gathered}\boxed{\bold{x=\frac{b_1c_2-b_2c_1}{a_1b_2-a_2b_1}}}\\\\\boxed{\bold{y=\frac{a_2c_1-a_1c_2}{a_1b_2-a_2b_1}}}\\\\\end{gathered}$

here

$\begin{gathered}a_1=1,b_1=1,c_1=-a-b\\\\a_2=a,b_2=-b,c_2=-a^2+b^2\\\\\end{gathered}$

Put these values in formula

$\begin{gathered}x=\frac{b_1c_2-b_2c_1}{a_1b_2-a_2b_1} \\ \\ x = \frac{ - {a}^{2} + {b}^{2} + b( - a - b)}{ - b - a} \\ \\ x = \frac{ - {a}^{2} + {b}^{2} - ab - {b}^{2} }{ - (a + b)} \\ \\ x = \frac{ - a(a + b)}{- (a + b)} \\ \\ \bold{x = a} \\ \end{gathered}$

Same way, put the coefficients to find y

$\begin{gathered}y=\frac{a_2c_1-a_1c_2}{a_1b_2-a_2b_1}\\\\y = \frac{ a( - a - b) -(- {a}^{2} + {b}^{2}) }{- (a + b)} \\ \\ y = \frac{ - {a}^{2} - ab + {a}^{2} - {b}^{2} }{- (a + b)} \\ \\ y = \frac{ - b(a + b)}{ - (a + b)} \\ \\\bold{y = b} \\ \\\end{gathered}$

Final answer:

$\begin{gathered}\bold{\red{x = a}} \\ \bold{\green{y = b}} \\ \end{gathered}$

Answer 2

Step-by-step explanation:

I hope this answer is correct.