Algebraic Methods of Solving a Pair of Linear Equations

Solve the following 

$2x+5y=1$

$3x+2y=7$:

Solve:

$\frac{x}{2}+y=0.8$ and $\frac{7}{x+{\displaystyle \frac{y}{2}}}=10$

Solve:

$\frac{11}{x}-\frac{7}{y}=1$ and $\frac{9}{x}-\frac{4}{y}=6$

Palace of Versailles' rectangular garden has a length of $48 \mathrm{m}$ and breadth of $32 \mathrm{m}$. It is divided into $4$ regions as shown below. It is known that $EB=DG=24 \mathrm{m}$ and $AH=FC=32 \mathrm{m}$. Given the area of region $EBFK=512 {\mathrm{m}}^2$, what is the area of region $HKGD$?

Solve $x+y= 2$and $x-y= 3$. What is the value of$x \text{and} y$?

Solve:

$4x+6y=14$

$9x-6y= 12$

Solve the simultaneous equation by elimination method: 

$4p+q+33=0$

$9p+q-21=0$

Solve $2x+3y=11$ and $x-2y=-12$ and hence find the value of $k$ for which $y=kx+9$.

Solve:

$\frac{3x}{2}-\frac{5y}{3}=-2$

$\frac{x}{2}+\frac{y}{2}=\frac{13}{6}$

If $kx + my + 4 = 0 \text{and} 2x+y+1 = 0$ the two equations are inconsistent, what is $k:m$?

Ruhi invested a certain amount of money in two schemes $A$ and $B$, which offer interest at the rate of $8\%$ per annum and $9\%$ per annum, respectively. She received $₹1860$ as annual interest. However had she interchanged the amount of investments in the two schemes, she would have received $₹20$ more as annual interest. How much money did she invest in each scheme?

If $x+y=50$ and $3x+y=90$ then the value of $x$ and $y$

$\frac{4}{x}-\frac{3}{y}=5$ and $\frac{3}{x}-\frac{4}{y}=2$, then $\frac{1}{x}-\frac{1}{y}=?$

If $p^4+q^4+p^2{q}^2=273$ and $p^2+q^2-pq=21,$ then one of the values of $\left(\frac{1}{p}+\frac{1}{q}\right)$ is:

For what value of $p$, the system of equations $2x+3y-14= 0$and $5x-py-14=0$ will not have a unique solution?

$\left|\begin{array}{cc}2& -y\\ 1& x\end{array}\right|=16$ and $\left|\begin{array}{cc}3& 2\\ y& x\end{array}\right|=3$. From the given determinants form two simultaneous equations and solve them.

Write a pair of linear equations in $2$ variables which have infinite solutions.

If a pair of linear equations given by $a_{1}x+b_{1}y+c_1=0 \text{and} a_{2}x+b_{2}y+c_2=0$ has a unique solution, then which of the following is true$?$

Solve:

$\frac{3}{7}x-\frac{5}{8}y= 33\frac{1}{3}$

$64x-17y= 12\frac{1}{2}$

Solve the following equations:

$$\begin{gathered} s-t=3 \\ \frac{s}{3}+\frac{t}{2}=6 \end{gathered}$$