Two linear equations in variables $x$ and $y$ are given below.

$a_{1}x+b_{1}y+c=0$

$a_{2}x+b_{2}y+c=0$

Which of the following pieces of information is independently sufficient to determine if a solution exists or not for this pair of linear equations?

I. $\frac{a_1}{b_1}=\frac{a_2}{b_2}=1$ II. $\frac{a_1}{a_2}=\frac{b_1}{b_2}$ III. $\frac{a_1}{a_2}=\frac{a_1}{b_1}\ne 1$ IV. $\frac{a_1}{a_2}\ne \frac{b_1}{b_2}$

Harsh correctly solved a pair of linear equations in two variables and found their only point of intersection as $\left(3, -2\right)$. One of the lines was $x-y=5$. Which of the following could have been the other line?

I: $3x-3y=15$

II: $2x-3y=12$

III: $2x-3y=14$

The sum of a father and his son is $36$ years. When the son reaches his father's present age, the sum of their ages will be $80$ years. What is the present age of the son?

Form a pair of linear equation for the following problem and find their solution by substitution method.

Five years hence, the age of Jacob will be three times that of his son. Five years ago, Jacob’s age was seven times that of his son. What are their present ages?

If $28x+17y=68$ and $17x+28y=67,$ find the value of $x+y$ and $x-y$

A shopkeeper sells a saree at $8\%$ profit and a sweater at $10\%$ discount, thereby, getting a sum Rs. $1008$. If she had sold the saree at $10\%$ profit and the sweater at $8\%$ discount, she would have got Rs. $1028$ then find the cost of the saree and the list price (price before discounted) of the sweater.