# Ex.3.1 Q3 Pair of Linear Equations in Two Variables Solution - NCERT Maths Class 10

## Question

The cost of \(2 \,\rm{kg}\) of apples and \(1 \,\rm{kg}\) of grapes on a day was found to be ₹ \(160\) After a month, the cost of \( 4\,\rm{kg}\) of apples and \( 2\,\rm{kg}\) of grapes is ₹ \(300\) Represent the situation algebraically and geometrically.

## Text Solution

**What is Known?**

(i) Cost of \(2 \,\rm{kg} \)of apples and \(1 \,\rm{kg} \) of grapes is ₹ \(160\)

(ii) Cost of \(4 \,\rm{kg} \) of apples and \(2 \,\rm{kg} \) of grapes is ₹ \(300.\)

**What is Unknown?**

Represent the situation geometrically and algebraically.

**Reasoning:**

Assuming the cost of \(1 \,\rm{kg} \) apples as ₹ *\(x\)* and the cost of \(1 \,\rm{kg} \) grapes as ₹ \(y,\) two linear equations can be formed for the above situation.

**Steps:**

Let the cost of \(1 \,\rm{kg} \) of apples be *\(x\)* and cost of \(1 \,\rm{kg} \) of grapes be *\(y\)*

Cost kg \(2 \,\rm{kg} \) apples and \(1 \,\rm{kg} \) of grapes is ₹ \(160.\)

Mathematically,

\[2x + y = 160\]

Also, cost kg \(4 \,\rm{kg} \) apples and \(2 \,\rm{kg} \) of grapes is ₹ \(300.\)

Mathematically,

\[\begin{align}4x + 2y &= 300\\2(2x + y) &= 300\\2x\, + \,y\,& = \,150\end{align}\]

Algebraic representation where *\(x\)* and *\(y\)* are the cost of \(1 \,\rm{kg} \) apple and \(1 \,\rm{kg} \) grapes respectively.

\[\begin{align}2x + y &= 160 \,\dots(1)\\2x + y& = 150\,\dots(2)\end{align}\]

Therefore, the algebraic representation is for equation \(1\) is:

\[\begin{align}2x + y &= 160\\y &= 160-2x\end{align}\]

And, the algebraic representation is for equation \(2\) is:

\[\begin{align}2x + y& = 150\\y &= 150-2x\end{align}\]

Let us represent these equations graphically. For this, we need at least two solutions for each equation. We give these solutions in table shown below.

\(x\) | \(50\) | \(30\) |

\(y = 160-2x\) | \(60\) | \(100\) |

\(x\) | \(30\) | \(40\) |

\(y = 150-2x\) | \(90\) | \(70\) |

The graphical representation is as follows.

Unit \(= 1\, \rm{cm} = \)₹ \(10\)

Since the lines are parallel hence no Solution