Question

$\huge\boxed{\fcolorbox{black}{black}{question}}$


For which values of a and b does the following pair of linear equations have an infinite number of solutions?

2x + 3y = 7

(a – b) x + (a + b) y = 3a + b – 2

ᴛʜᴀɴᴋ ᴍʏ ᴀɴsᴡᴇʀ❥︎​

Answer 1

$\huge\red{A}\pink{N}\orange{S}\green{W}\blue{E}\gray{R:-}$

For 2x + 3y = 7, we have a1 = 2, b1 = 3, c1 = - 7 and

for (a - b) x + (a + b) y = 3a + b - 2,

we have a2 = a - b, b2 = a + b, c2 = - (3a + b - 2) .

$For \: infinite \: number \: of \: solutions \\ \frac{a1}{a2} = \frac{ b 1 }{b2} = \frac{c1}{c2}$

$\longmapsto\tt \frac{2}{a - b} = \frac{3}{a + b} = \frac{ - 7}{ - (3a + b - 2)}$

$\small\red{consider \: \frac{2}{a - b} = \frac{3}{a + b} }$

$\small\pink{=>2a + 2b = 3a - 3b}$

$\small\orange{ = > a = 5b \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: ➡ (1) }$

$\small\green{and \: \frac{2}{a - b} = \frac{7}{3a + b - 2} }$

$\small\blue{=> 6a + 2b - 4 = 7a - 7b}</p><p>$

$\small\blue{ = > - a + 9b = 4}$

$\small\blue{ = > - \: 5b + 9b = 4}$

$\small\blue{=> 4b = 4 \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: [from (1) ]}$

$\small\gray{From (1), a = 5 × 1 = 5}$

∴ For a = 5, b = 1, the pair of equations has infinitely many solutions.