Introduction to Pair of Linear Equations in Two Variables

The ratio of a two-digit number $:$ sum of its digits is $7:1.$ How many such two-digit numbers are possible$?$

If $5x=y-10$ is expressed in its standard form as $ax+by+c=0$, then the value of $c$ is

The perimeter of rectangle $ABCD=40 \mathrm{cm}$. The sum of the perimeters of rectangle $AEFD$ and rectangle $BEFC$ is $20\%$ more than that of $ABCD$. The area (in ${\mathrm{cm}}^2$) of the rectangle $ABCD$ is

A pair of linear equations can be consistent but cannot be inconsistent

A pair of linear equations which is consistent can be dependent but not Independent.

A pair of linear equations which is consistent can be dependent or Independent

A pair of linear equations can be consistent or inconsistent

Two trains $100 \mathrm{m}$ and $80 \mathrm{m}$ long is running along the parallel rail lines. When they run in the same direction, then the train of faster speed crosses the train of slower speed in $18 \mathrm{seconds}$. And when they run in the opposite direction with equal speed, then they meet in $9 \mathrm{seconds}$. Find the speed of the two trains. 

In a given fraction, if $1$ is subtracted from the numerator and $2$ is added to the denominator, it becomes $\frac{1}{2}$. If $7$ is subtracted from the numerator and $2$ is subtracted from the denominator, it becomes $\frac{1}{3}$. Then the fraction is

The area of a trapezium is $1400 {\mathrm{cm}}^2$. Its altitude is $50 \mathrm{cm}$. Find the two bases if the number of $\mathrm{cms}$ in each base is an integer divisible by $9$. The number of solution to this problem is

By running $900 \mathrm{metres}$ and walking another $900 \mathrm{metres}$, a man covers a distance of $1800 \mathrm{m}$ in$10\frac{1}{2}$ minutes. If he had run $20\%$ slower and walked $20\%$ faster, he would have taken just $10 \mathrm{minutes}$ to cover the distance. Find the time taken by him to walk $100 \mathrm{m}.$

By running $900 \mathrm{metres}$ and walking another $\mathit{900}\mathit{ }metres$, a man covers a distance of $1800 \mathrm{m}$ in$10\frac{1}{2}$ minutes. If he had run $20\%$ slower and walked $20\%$ faster, he would have taken just $10 \mathrm{minutes}$ to cover the distance. Find the time taken by him to run $100 \mathrm{m}$.

The students of a class are made to stand in rows. If number of students in each row is reduced by $2$, the number of rows increases by $1$ but if the number of students in each row is increased by $3$, the number of rows is reduced by $1$. Find the number of students in the class.

A person on tour has $\mathrm{Rs}. 360$ for his daily expenses. If he extends his tour by $4 \mathrm{days}$, he has to cut down his daily expenses by $\mathrm{Rs}. 3$. Find the original duration of the tour.

$2$ tables and $3$ chairs together cost $\mathrm{Rs}. 2000$ where as $3$ tables and $2$ chairs together cost $\mathrm{Rs}. 2500$. Find the total cost of $1$ table and $5$ chairs.

Frame a formula for the following statement: The Fahrenheit temperature $F$, decreased by $32$ is equal to nine-fifth of the Centigrade temperature $C$. Given $C=40°$, find $F$ in Kelvin.

The age of a man is three times the sum of the ages of his two children and $5$ years hence his age will be double the sum of their ages. Find his present age.

If the length and the breadth of a room are increased by $1$ metre, the area is increased by $21$ square metres. If the length is increased by $1$ metre and breadth is decreased by $1$ metre, the area is decreased by $5$ square metres. Find the perimeter of the room.

If $1$ is added to the denominator of a fraction, it becomes $\frac{1}{2}$. If $1$ is added to the numerator of the fraction, it becomes $1$. Find the fraction.

If the numerator of a fraction is increased by $2$ and the denominator by $1$, it becomes $\frac{5}{8}$ and if the numerator and the denominator of the same fraction are each increased by $1$, the fraction becomes $\frac{1}{2}$. Find the fraction.