# 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):

**(5.4)**

If we multiply the first of these equations by ${b}_{2}$ and the second by ${b}_{1}\hspace{0.17em},\hspace{0.17em}$ we obtain

**(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

**(5.6)**

In the same manner, we may show that

**(5.7)**

The expression ${a}_{1}{b}_{2}\hspace{0.17em}-\hspace{0.17em}{a}_{2}{b}_{1}\hspace{0.17em},\hspace{0.17em}$ which appears in each of the denominators of Eqs. (5.6)

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