Pair of Linear Equations in Two Variables | Exercise 3.1

Question 1

Form the pair of linear equations in the following problems, and find their solutions graphically.

(i) 10 students of Class X took part in a Mathematics quiz. If the number of girls is 4 more than the number of boys, find the number of boys and girls who took part in the quiz.

(ii) 5 pencils and 7 pens together cost ₹ 50, whereas 7 pencils and 5 pens together cost ₹ 46. Find the cost of one pencil and that of one pen.

Solution

Understand the Question

To solve word problems graphically using a pair of linear equations:
1. Represent the unknown quantities as two variables, $x$ and $y$ .
2. Formulate two linear equations based on the given conditions.
3. Find at least two coordinate pairs $(x, y)$ satisfying each linear equation.
4. Plot the corresponding lines on a Cartesian plane; the coordinates of their point of intersection $(x, y)$ give the unique solution.

(i) 10 students of Class X took part in a Mathematics quiz. If the number of girls is 4 more than the number of boys, find the number of boys and girls who took part in the quiz.

Step 1 · Formulate Linear Equations

Let the number of boys be $x$ and the number of girls be $y$ .

Total number of students is $10$ : $x + y = 10 \quad \dots (1)$

The number of girls is $4$ more than the number of boys:

\begin{aligned} y &= x + 4 \\ y - x &= 4 \quad \dots (2) \end{aligned}

Step 2 · Find Points for Equation (1)

For $x + y = 10 \implies y = 10 - x$ :

If $x = 0$ :

\begin{aligned} 0 + y &= 10 \\ y &= 10 \end{aligned}

Point: $(0, 10)$

If $y = 0$ :

\begin{aligned} x + 0 &= 10 \\ x &= 10 \end{aligned}

Point: $(10, 0)$

If $x = 5$ :

\begin{aligned} 5 + y &= 10 \\ y &= 10 - 5 = 5 \end{aligned}

Point: $(5, 5)$

Step 3 · Find Points for Equation (2)

For $y - x = 4 \implies y = x + 4$ :

If $x = 0$ :

\begin{aligned} y - 0 &= 4 \\ y &= 4 \end{aligned}

Point: $(0, 4)$

If $y = 0$ :

\begin{aligned} 0 - x &= 4 \\ x &= -4 \end{aligned}

Point: $(-4, 0)$

If $x = 3$ :

\begin{aligned} y - 3 &= 4 \\ y &= 4 + 3 = 7 \end{aligned}

Point: $(3, 7)$

Step 4 · Plot Graph and Find Intersection

Diagram 1

Plotting the lines on the graph, they intersect at the point $(3, 7)$ .

Therefore, $x = 3$ and $y = 7$ .

Answer

(i) Number of boys $= 3$ , Number of girls $= 7$

(ii) 5 pencils and 7 pens together cost ₹ 50, whereas 7 pencils and 5 pens together cost ₹ 46. Find the cost of one pencil and that of one pen.

Step 1 · Formulate Linear Equations

Let the cost of one pencil be ₹ $x$ and the cost of one pen be ₹ $y$ .

Cost of $5$ pencils and $7$ pens is ₹ $50$ : $5x + 7y = 50 \quad \dots (3)$

Cost of $7$ pencils and $5$ pens is ₹ $46$ : $7x + 5y = 46 \quad \dots (4)$

Step 2 · Find Points for Equation (3)

For $5x + 7y = 50 \implies y = \dfrac{50 - 5x}{7}$ :

If $x = 3$ :

\begin{aligned} 5(3) + 7y &= 50 \\ 15 + 7y &= 50 \\ 7y &= 50 - 15 = 35 \\ y &= 5 \end{aligned}

Point: $(3, 5)$

If $x = 10$ :

\begin{aligned} 5(10) + 7y &= 50 \\ 50 + 7y &= 50 \\ 7y &= 0 \\ y &= 0 \end{aligned}

Point: $(10, 0)$

Step 3 · Find Points for Equation (4)

For $7x + 5y = 46 \implies y = \dfrac{46 - 7x}{5}$ :

If $x = 3$ :

\begin{aligned} 7(3) + 5y &= 46 \\ 21 + 5y &= 46 \\ 5y &= 46 - 21 = 25 \\ y &= 5 \end{aligned}

Point: $(3, 5)$

If $x = 8$ :

\begin{aligned} 7(8) + 5y &= 46 \\ 56 + 5y &= 46 \\ 5y &= 46 - 56 = -10 \\ y &= -2 \end{aligned}

Point: $(8, -2)$

Step 4 · Plot Graph and Find Intersection

Plotting the lines on the graph, they intersect at the common point $(3, 5)$ .

Therefore, $x = 3$ and $y = 5$ .

Answer

(ii) Cost of one pencil $= \text{₹ } 3$ , Cost of one pen $= \text{₹ } 5$

Common Mistakes

Interchanging Variables: Confusing which variable represents which quantity (e.g., mixing up the number of boys and girls, or pencil and pen costs).
Non-integer Coordinate Values: Picking values of $x$ that produce fractions or repeating decimals for $y$ , making graphing difficult and inaccurate. Always look for integer solutions where possible.
Reading the Solution: Forgetting that the solution to a pair of linear equations is the intersection point $(x, y)$ of both lines on the graph.

Question 1

More questions in Exercise 3.1