Question

Two sides of a parallelogram are contained in lines whose equations are $2.3x+1.2y=2.1$and $4.1x-0.5y=14.3$. Find the coordinates of a vertex of the parallelogram.

Short answer

The required coordinates are $\left(3,-4\right)$.

Step-by-step solution

Step 1- Determine the value of x

Use the provided equations and apply the Cramer’s rule for the value of x.

$x=\frac{\left|\begin{array}{cc}e& b\\ f& d\end{array}\right|}{\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|}$

Substitute $2.3$ in place of a, $4.1$ in place of c, $1.2$ in place of b, $-0.5$ in place of d, $2.1$ in place of e and role="math" localid="1647245624304" $14.3$ in place of f into role="math" localid="1647245617928" $x=\frac{\left|\begin{array}{cc}e& b\\ f& d\end{array}\right|}{\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|}$and perform simplification of determinants to write the value of x.

$\begin{array}{c}x=\frac{\left|\begin{array}{cc}2.1& 1.2\\ 14.3& -0.5\end{array}\right|}{\left|\begin{array}{cc}2.3& 1.2\\ 4.1& -0.5\end{array}\right|}\\ =\frac{\left(2.1\right)\cdot \left(-0.5\right)-\left(14.3\right)\cdot \left(1.2\right)}{2.3\cdot \left(-0.5\right)-\left(4.1\right)\cdot \left(1.2\right)}\\ =\frac{-18.21}{-6.07}\\ =3\end{array}$

– Determine the value of y 

Apply the Cramer’s rule for the value of y.

$y=\frac{\left|\begin{array}{cc}a& e\\ c& f\end{array}\right|}{\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|}$

Substitute $2.3$ in place of a, $4.1$ in place of c, $1.2$ in place of b, $-0.5$ in place of d, $2.1$ in place of e and role="math" localid="1647245803291" $14.3$ in place of f into role="math" localid="1647245798680" $y=\frac{\left|\begin{array}{cc}a& e\\ c& f\end{array}\right|}{\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|}$and perform simplification of determinants to write the value of y.

$\begin{array}{c}y=\frac{\left|\begin{array}{cc}2.3& 2.1\\ 4.1& 14.3\end{array}\right|}{\left|\begin{array}{cc}2.3& 1.2\\ 4.1& -0.5\end{array}\right|}\\ =\frac{2.3\cdot \left(14.3\right)-\left(4.1\right)\cdot 2.1}{2.3\cdot \left(-0.5\right)-\left(4.1\right)\cdot \left(1.2\right)}\\ =\frac{24.28}{-6.07}\\ =-4\end{array}$

Step 3- Write the solution

The obtained value of x is 3 and y is $-4$ so the coordinates are $\left(3,-4\right)$.