Introduction to Pair of Linear Equations in Two Variables

Arpit can friends a garden in $3$ hours, Ankur can do it in $5$ hours and Akash take $6$ hours to do the same if they work together in how many hours can they fence the garden?

A test consists of 'True' or 'False' questions. One mark is awarded for every correct answer while $\frac{1}{4}$ mark is deducted for every wrong answer. A student knew the answers to some of the questions. Rest of the questions he attempted by guessing. He answered $120$ questions and got $90$ marks. 

If answers to all questions he attempted by guessing were wrong then how many questions did he answer correctly?

Solve:

$\frac{4}{3}x+2y=8; 2x+3y=12$

Anuj had some chocolates, and he divided them into two lots $\mathrm{A}$ and $\mathrm{B}$. He sold the first lot at the rate of $₹2$ form $3$ chocolates and the second lot at the ratio $₹1$ per chocolate, and got a total of $₹400$. If he had sold the first lot at the rate of $₹1$ per chocolate, and the second lot at the rate of $₹4$ for $5$ chocolates, his total collection would have been $₹460$. Find the total number of chocolates he had.

Shown below is a graph with four straight lines. It is given that $k_1, k_2$ and $k_3$ intersect at exactly one point and line $k_3\parallel k_4$

Which of the following is the equation of the line $k_3$?

Given below are two equations.

${49}^{\left(x+3y\right)}=7$; $7^{\left(4x+12y\right)}={49}^4$

Frame a pair of linear equations in two variables by simplifying the given equations. Do these linear equations form a pair of coincident lines?

Answer the question based on the given information.

The total cost of snowden ice cream parlour is divided into fixed cost $\left(x\right)$ and variable cost $\left(y\right).$ Fixed cost is the cost that the ice cream parlour has to incur even at zero level of production and variable cost is the cost that will be directly proportional to each unit of ice cream sold.

The parlour launched a new flavour of ice cream and wanted to find the fixed and variable cost associated with it. They found that their total cost for that flavour was Rs $27500$ after selling $150$ units and Rs $32500$ after selling $250$ units.

Frame the equations that represent the total cost incurred by snowden ice cream parlour for the new flavor in terms of fixed and variable costs.

Given below is a non-linear equation.

$\frac{3\left(x+y\right)+7\left(x-y\right)}{x^2-y^2}=12$

Garima has reduced it to a linear equation in two variables. Shown below are her steps.

Step $1:$ $\frac{3\left(x+y\right)+7\left(x-y\right)}{x^2-y^2}=12$

Step $2: \frac{3}{\left(x+y\right)}+\frac{7}{\left(x-y\right)}=12$

Step $3:$Takes $\frac{1}{\left(x+y\right)}=a$ and $\frac{1}{\left(x-y\right)}=b$

Step $4: 3a+7b=12$

However, she has made a mistake in one step. Identify the erroneous step and complete the solution.

The two circles represent the ordered pairs $\left(a,b\right),$ which are solutions of the respective equations. The circles are divided into $3$ regions $P,Q$ and $R$ as shown.

Write one ordered pair each belonging to $P,Q$ and $R.$ SHow your work.

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

The difference between two numbers is $10$. If one of the number is double the other number, find both the numbers.

A plane flying north at $500 mph$ passes over a city at $12$noon. A plane flying east at the sea altitude passes over the same city at $12:30$ P.M. The plane is flying east at $400 mph$. To the nearest hundred miles, how far apart the two planes at $2$ P.M.?

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

$\sqrt{x}+y=1$
$x+\sqrt{y}=2$
Find values of $x$ and $y$.

When I was $\frac{3}{5}$ th of your present age, you were $\frac{1}{3}$ rd of my present age. If my present age after $4$ years will be $40$, then after how many years will your age be $40?$

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

Half the perimeter of a rectangular garden, whose length is $4 \mathrm{m}$ more than its width, is $36 \mathrm{m}$. Find the dimensions of the garden.

Explain linear equations?

$P,Q \text{and} R$ together have $180$ candies among them. $P$ gives $Q$ and $R$ each as many candies as they already have. After this, $R$ gives $Q$ half as many candies as $Q$ already has, and $R$ also gives $P$ twice as many candies as $P$ already has. Now each of them has the same number of candies with them. What is the ratio of the respective number of candies of $P$ and $R$. 

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