Basic Technical Mathematics, 11th Edition

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):

$$\begin{array}{l}{a}_{1}x\hspace{0.17em}+\hspace{0.17em}{b}_{1}y\hspace{0.17em}=\hspace{0.17em}{c}_1\\ a_{2}x\hspace{0.17em}+\hspace{0.17em}{b}_{2}y\hspace{0.17em}=\hspace{0.17em}{c}_2\end{array}$$(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

$$\begin{array}{l}{a}_1{b}_{2}x\hspace{0.17em}+\hspace{0.17em}{b}_1{b}_{2}y\hspace{0.17em}=\hspace{0.17em}{c}_1{b}_2\\ a_2{b}_{1}x\hspace{0.17em}+\hspace{0.17em}{b}_2{b}_{1}y\hspace{0.17em}=\hspace{0.17em}{c}_2{b}_1\end{array}$$(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

$$x\hspace{0.17em}=\hspace{0.17em}{\displaystyle \frac{c_1{b}_2\hspace{0.17em}-\hspace{0.17em}{c}_2{b}_1}{a_1{b}_2\hspace{0.17em}-\hspace{0.17em}{a}_2{b}_1}}$$(5.6)

In the same manner, we may show that

$$y\hspace{0.17em}=\hspace{0.17em}{\displaystyle \frac{a_1{c}_2\hspace{0.17em}-\hspace{0.17em}{a}_2{c}_1}{a_1{b}_2\hspace{0.17em}-\hspace{0.17em}{a}_2{b}_1}}$$(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)