Question

Two sides of an angle are contained in lines whose equations are $4x+y=-4$and $2x-3y=-9$. Find the coordinates of the vertex of the angle.

Short answer

The required solution is $\left(-\frac{3}{2},2\right)$.

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 4 in place of a, 2 in place of c, 1 in place of b, $-3$ in place of d, $-4$ in place of e and role="math" localid="1647244918633" $-9$ in place of f into role="math" localid="1647244913479" $x=\frac{\left|\begin{array}{cc}e& b\\ f& d\end{array}\right|}{\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|}$and perform simplification of determinants to write the value of x.

$\begin{array}{c}x=\frac{\left|\begin{array}{cc}-4& 1\\ -9& -3\end{array}\right|}{\left|\begin{array}{cc}4& 1\\ 2& -3\end{array}\right|}\\ =\frac{\left(-4\right)\cdot \left(-3\right)-\left(1\right)\cdot \left(-9\right)}{4\cdot \left(-3\right)-\left(1\right)\cdot 2}\\ =\frac{21}{-14}\\ =-\frac{3}{2}\end{array}$

– 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 4 in place of a, 2 in place of c, 1 in place of b, $-3$ in place of d, $-4$ in place of e and role="math" localid="1647245081200" $-9$ in place of f into role="math" localid="1647245075902" $y=\frac{\left|\begin{array}{cc}a& e\\ c& f\end{array}\right|}{\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|}$and perform simplification of determinants to write the value of y.

$\begin{array}{c}y=\frac{\left|\begin{array}{cc}4& -4\\ 2& -9\end{array}\right|}{\left|\begin{array}{cc}4& 1\\ 2& -3\end{array}\right|}\\ =\frac{4\cdot \left(-9\right)-\left(-4\right)\cdot 2}{4\cdot \left(-3\right)-\left(1\right)\cdot 2}\\ =\frac{-28}{-14}\\ =2\end{array}$

Step 3- Write the solution

The obtained value of x is $-\frac{3}{2}$ and y is 2 so the coordinates are $\left(-\frac{3}{2},2\right)$.