Question

If 2a+3b=12 and a−3b=0, then what is the value of a?
If 7a+3b=20 and 3a+7b=10, then what is the value of a+b?
If a+b=12 and a−b = 8, what is the value of a2+b2?
If 8a−4b=20 and 4a−8b=4, what is the value of a−b?

Answer 1

$\textbf{Given:}$

$(i)\;7a+3b=20\;\text{and}\;3a+7b=10$

$(ii)\;a+b=12\;\text{and}\;a-b=8$

$(iii)\;8a-4b=20\;\text{and}\;4a-8b=4$

$\textbf{Solution:}$

$(i)\bf\;7a+3b=20\;\textbf{and}\;3a+7b=10$

$\text{Adding these equations}$

$10a+10b=30$

$\implies\boxed{\bf\,a+b=3}$......(1)

$\text{subtracting the given equations, we get}$

$4a-4b=10$

$\implies\,a-b=\frac{5}{2}$....(2)

$\text{Adding (1) and (2), we get}$

$2a=\frac{11}{2}$

$\implies\boxed{\bf\,a=\frac{11}{4}}$

$\bf(ii)\;a+b=12\;\textbf{and}\;a-b=8$

$\text{Adding these equations, we get}$

$2a=20$

$\implies\bf\,a=10$

$a+b=12\implies\bf\,b=2$

$a^2+b^2=10^2+^2=100+4=104$

$\implies\boxed{\bf\,a^2+b^2=104}$

$\bf(iii)\;8a-4b=20\;\text{and}\;4a-8b=4$

$\text{Adding these equations, we get}$

$12a-12b=24$

$\implies\boxed{\bf\,a-b=2}$

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