5.4 Solving Systems of Two Linear Equations in Two Unknowns by Determinants

  • Determinant of the Second Order • Cramer’s Rule • Solving Systems of Equations by Determinants

Consider two linear equations in two unknowns, as given in Eqs. (5.4):

a1x + b1y = c1a2x + b2y = c2(5.4)

If we multiply the first of these equations by b2 and the second by b1 ,  we obtain

a1b2x + b1b2y = c1b2a2b1x + b2b1y = c2b1(5.5)

We see that the coefficients of y are the same. Thus, subtracting the second equation from the first, we can solve for x. The solution can be shown to be

x = c1b2 − c2b1a1b2 − a2b1(5.6)

In the same manner, we may show that

y = a1c2 − a2c1a1b2 − a2b1(5.7)

The expression a1b2 − a2b1 ,  which appears in each of the denominators of Eqs. (5.6)

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