Question

Find the value of alpha and beta for which the following pair of linear equations has infinite number of solutions.
2x + 3y = 7.
2 alpha x + ( alpha + beta ) y = 28.

Answer 1

Dear Student,

For two lines to have infinite solutions the two lines should be parallel to each other.

For equation: ax + by + c = 0
Slope = $-\frac{\mathrm{a}}{\mathrm{b}}$
For line : 2x + 3y = 7

Slope = $-\frac{2}{3}$
For line : (2 alpha) x + (alpha+beta)y = 28
Slope = $-\frac{2\mathrm{alpha}}{\mathrm{alpha}+\mathrm{beta}}$
Slope of two line should be equal, therefore
$-\frac{2}{3}$ = $-\frac{2\mathrm{alpha}}{\mathrm{alpha}+\mathrm{beta}}$
$\frac{1}{3}$ = $\frac{\mathrm{alpha}}{\mathrm{alpha}+\mathrm{beta}}$
Therefore, alpha = 1 and beta = 2

Regards