Solve the system for x and y using Cramer's rule. Assume a and b are nonzero constants.
$\begin{array} { l }
x + a y = \frac { 1 } { a } \
x + b y = \frac { 1 } { b }
\end{array}$
A) $\left\{ \left( b + a , - \frac { 1 } { a b } \right\} \right\}$
B) $\left\{ \left( \frac { 1 } { a b } , \frac { a - b } { a b } \right\} \right\}$
C) $\left\{ \left( \frac { a + b } { a b } , - \frac { 1 } { a b } \right) \right\}$
D) $\left\{ \left( \frac { b - a } { a b } , - \frac { 1 } { a b } \right) \right\}$

Question

Solve the system for x and y using Cramer's rule. Assume a and b are nonzero constants.
$\begin{array} { l } 
x + a y = \frac { 1 } { a } \
x + b y = \frac { 1 } { b }
\end{array}$
A) $\left\{ \left( b + a , - \frac { 1 } { a b } \right\} \right\}$
B) $\left\{ \left( \frac { 1 } { a b } , \frac { a - b } { a b } \right\} \right\}$
C) $\left\{ \left( \frac { a + b } { a b } , - \frac { 1 } { a b } \right) \right\}$
D) $\left\{ \left( \frac { b - a } { a b } , - \frac { 1 } { a b } \right) \right\}$

Quizplus · Accepted Answer

Cramer's rule involves calculating determinants to solve for variables. For the given system:$$\begin{align*}x + a y &= \frac { 1 } { a } \x + b y &= \frac { 1 } { b }\end{align*}$$The determinant of the coefficient matrix ($D$) is:$$D = \begin{vmatrix}1 & a \1 & b\end{vmatrix} = 1(b) - 1(a) = b - a$$The determinant for $x$ ($D_x$) is:$$D_x = \begin{vmatrix}\frac{1}{a} & a \\frac{1}{b} & b\end{vmatrix} = \frac{1}{a}b - \frac{1}{b}a = \frac{b}{a} - \frac{a}{b} = \frac{b^2 - a^2}{ab}$$The determinant for $y$ ($D_y$) is:$$D_y = \begin{vmatrix}1 & \frac{1}{a} \1 & \frac{1}{b}\end{vmatrix} = 1\left(\frac{1}{b}\right) - 1\left(\frac{1}{a}\right) = \frac{1}{b} - \frac{1}{a} = \frac{a - b}{ab}$$Solving for $x$ and $y$:$$x = \frac{D_x}{D} = \frac{\frac{b^2 - a^2}{ab}}{b - a} = \frac{b + a}{ab}$$$$y = \frac{D_y}{D} = \frac{\frac{a - b}{ab}}{b - a} = -\frac{1}{ab}$$Thus, the solution is $x = \frac{a + b}{ab}$, $y = -\frac{1}{ab}$, which matches choice C.

Solve the System for X and Y Using Cramer's Rule x+ay=1ax+by=1b\begin{array} { l } x + a y = \frac { 1 } { a } \\x + b y = \frac { 1 } { b }\end{array}x+ay=a1​x+by=b1​​

Solve the System for X and Y Using Cramer's Rule $\begin{array} { l } x + a y = \frac { 1 } { a } \\x + b y = \frac { 1 } { b }\end{array}$