# Using the determinants given below form two linear equations and solve them D = |572-3|, Dy = |542-10| - Algebra

Sum

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

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

#### Solution

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

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

Dx = |("c"_1, "b"_1),("c"_2, "b"_2)|

Dy = |("a"_1, "c"_1),("a"_2, "c"_2)|   ......(i)

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

Dy = |(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.

Concept: Determinant of Order Two
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