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Q1(ii):
Form the pair of linear equations in the following problems, and find their solutions graphically. (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 :
Given:
1. The cost of 5 pencils and 7 pens together is ₹ 50.
2. The cost of 7 pencils and 5 pens together is ₹ 46.
To Find:
The cost of one pencil and the cost of one pen.
Step 1: Defining Variables
Let the cost of one pencil be $x$ and the cost of one pen be $y$.
Step 2: Formulating the Equations
Based on the given conditions, we can write the following system of linear equations:
Equation 1: $5x + 7y = 50$
Equation 2: $7x + 5y = 46$
Step 3: Finding Coordinates for Graphical Representation
To plot these lines, we find at least two points for each equation.
For Equation 1: $5x + 7y = 50 \implies y = \frac{50 - 5x}{7}$
| $x$ | $y$ |
|---|---|
| 3 | 5 |
| 10 | 0 |
For Equation 2: $7x + 5y = 46 \implies y = \frac{46 - 7x}{5}$
| $x$ | $y$ |
|---|---|
| 3 | 5 |
| 8 | -2 |
Step 4: Visual Representation
Step 5: Solving the System
From the table of values, we observe that the point $(3, 5)$ satisfies both equations.
Verification for Equation 1: $5(3) + 7(5) = 15 + 35 = 50$ (Correct)
Verification for Equation 2: $7(3) + 5(5) = 21 + 25 = 46$ (Correct)
Step 6: Conclusion
Since the lines intersect at the point $(3, 5)$, the solution to the system is $x = 3$ and $y = 5$.
Final Answer: The cost of one pencil is ₹ 3 and the cost of one pen is ₹ 5.
More Questions from Class 10 Mathematics Pair of linear equations in two variable EXERCISE 3.1
- Q1(i): 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.
- Q2(i): On comparing the ratios $\frac{a_1}{a_2}$, $\frac{b_1}{b_2}$, and $\frac{c_1}{c_2}$, find out whether the lines representing the following pairs of linear equations intersect at a point, are parallel or coincident: (i) 5x – 4y + 8 = 0; 7x + 6y – 9 = 0
- Q2(ii): On comparing the ratios $\frac{a_1}{a_2}$, $\frac{b_1}{b_2}$, and $\frac{c_1}{c_2}$, find out whether the lines representing the following pairs of linear equations intersect at a point, are parallel or coincident: (ii) 9x + 3y + 12 = 0; 18x + 6y + 24 = 0
- Q2(iii): On comparing the ratios $\frac{a_1}{a_2}$, $\frac{b_1}{b_2}$, and $\frac{c_1}{c_2}$, find out whether the lines representing the following pairs of linear equations intersect at a point, are parallel or coincident: (iii) 6x – 3y + 10 = 0; 2x – y + 9 = 0
- Q3(i): On comparing the ratios $\frac{a_1}{a_2}$, $\frac{b_1}{b_2}$, and $\frac{c_1}{c_2}$, find out whether the following pair of linear equations are consistent, or inconsistent. (i) 3x + 2y = 5 ; 2x – 3y = 7
- Q3(ii): On comparing the ratios $\frac{a_1}{a_2}$, $\frac{b_1}{b_2}$, and $\frac{c_1}{c_2}$, find out whether the following pair of linear equations are consistent, or inconsistent. (ii) 2x – 3y = 8 ; 4x – 6y = 9
- Q3(iii): On comparing the ratios $\frac{a_1}{a_2}$, $\frac{b_1}{b_2}$, and $\frac{c_1}{c_2}$, find out whether the following pair of linear equations are consistent, or inconsistent. (iii) $\frac{3}{2}x + \frac{5}{3}y = 7$ ; 9x – 10y = 14
- Q3(iv): On comparing the ratios $\frac{a_1}{a_2}$, $\frac{b_1}{b_2}$, and $\frac{c_1}{c_2}$, find out whether the following pair of linear equations are consistent, or inconsistent. (iv) 5x – 3y = 11 ; – 10x + 6y = –22
- Q3(v): On comparing the ratios $\frac{a_1}{a_2}$, $\frac{b_1}{b_2}$, and $\frac{c_1}{c_2}$, find out whether the following pair of linear equations are consistent, or inconsistent. (v) $\frac{4}{3}x + 2y = 8$ ; 2x + 3y = 12
- Q4(i): Which of the following pairs of linear equations are consistent/inconsistent? If consistent, obtain the solution graphically: (i) x + y = 5, 2x + 2y = 10
- Q4(ii): Which of the following pairs of linear equations are consistent/inconsistent? If consistent, obtain the solution graphically: (ii) x – y = 8, 3x – 3y = 16
- Q4(iii): Which of the following pairs of linear equations are consistent/inconsistent? If consistent, obtain the solution graphically: (iii) 2x + y – 6 = 0, 4x – 2y – 4 = 0
- Q4(iv): Which of the following pairs of linear equations are consistent/inconsistent? If consistent, obtain the solution graphically: (iv) 2x – 2y – 2 = 0, 4x – 4y – 5 = 0
- Q5: Half the perimeter of a rectangular garden, whose length is 4 m more than its width, is 36 m. Find the dimensions of the garden.
- Q6(i): Given the linear equation 2x + 3y – 8 = 0, write another linear equation in two variables such that the geometrical representation of the pair so formed is: (i) intersecting lines
- Q6(ii): Given the linear equation 2x + 3y – 8 = 0, write another linear equation in two variables such that the geometrical representation of the pair so formed is: (ii) parallel lines
- Q6(iii): Given the linear equation 2x + 3y – 8 = 0, write another linear equation in two variables such that the geometrical representation of the pair so formed is: (iii) coincident lines
- Q7: Draw the graphs of the equations x – y + 1 = 0 and 3x + 2y – 12 = 0. Determine the coordinates of the vertices of the triangle formed by these lines and the x-axis, and shade the triangular region.
CBSE Solutions for Class 10 Mathematics Pair of linear equations in two variable
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