Using the determinants given below form two linear equations and solve them

D = `|(5, 7),(2, -3)|`, D_{y} = `|(5, 4),(2,-10)|`

#### Solution

If a_{1}x + b_{1}y = c_{1} and a_{2}x + b_{2}y = c_{2} are linear equations in two variables, then

D = `|("a"_1, "b"_1),("a"_2, "a"_2)|`

D_{x} = `|("c"_1, "b"_1),("c"_2, "b"_2)|`

D_{y} = `|("a"_1, "c"_1),("a"_2, "c"_2)|` ......(i)

Given, D = `|(5, 7),(2, -3)|`

D_{y} = `|(5, 4),(2, -10)|`

Comparing these determinants with equation (i), we get

a_{1} = 5, b_{1} = 7, c_{1} = 4

a_{2} = 2, b_{2} = –3, c_{2} = –10

∴ The equations are

5x + 7y = 4 ......(ii)

2x– 3y = –10 ......(iii)

Multiplying equation (ii) by 3 and equation (iii) by 7, we get

15x + 21y = 12 ......(iv)

14x – 21y = –70 ......(v)

Adding equations (iv) and (v), we get

15x + 21y = 12

+ 14x – 21y = – 70

29x = – 58

∴ x = `-58/29` = – 2

Substituting x = –2 in equation (ii), we get

5(–2) + 7y = 4

∴ –10 + 7y = 4

∴ 7y = 14

∴ y = `14/7` = 2

∴ x = –2 and y = 2 is the solution of the equations 5x + 7y = 4 and 2x – 3y = –10.