Question

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

D = `|(5, 7),(2, -3)|`, Dy = `|(5, 4),(2, -10)|`

Answer 1

If a₁x + b₁y = c₁ and a₂x + b₂y = c₂ are linear equations in two variables, then

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

a₁ = 5, b₁ = 7, c₁ = 4

a₂ = 2, b₂ = –3, c₂ = –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.