Exercise 3.1 (NCERT) – Pair of Linear Equations in Two Varibales, Class 10
| Chapter 3 | Exercise 3.1 | Class 10 |
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Below are the solutions of every question of exercise 3.1 of chapter 3, Pair of Linear Equations in Two Variables, NCERT of class 10. YouTube video tutorial of every question is also embedded along with the written solution.
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.
Total students = 10
Let the No. of boys be x
Then the No. of girls = x + 4
(ii) 5 pencils and 7 pens together cost Rs. 50, whereas 7 pencils and 5 pens together cost Rs. 46. Find the cost of one pencil and that of one pen.
Let the cost of one pencil be x
and the cost of one pen be y
Cost of 5 pencils and 7 pens = 50
=> 5x + 7y = 50 … (1)
Cost of 7 pencils and 5 pens = 46
=> 7x + 5y = 46 … (2)
Multiply (1) by 5 and (2) by 7 to eliminate y
Subtract (3) from (4)
Substitute x = 3 in (1)
Cost of 1 Pencil = Rs. 3
Cost of 1 Pen = Rs. 5
Question 2
On comparing the ratios a1/a2, b1/b2 and c1/c2 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
Comparing the equations with standard form of Linear Equations in Two variables:
a1x + b1y + c1 = 0 &
a2x + b2y + c2 = 0
a1 = 5, b1 = -4, c1 = 8
a2 = 7, b2 = 6, c2 = -9
Lines intersect each other at one point, Unique Solution
(ii) 9x + 3y + 12 = 0
18x + 6y + 24 = 0
Comparing the equations with standard form of Linear Equations in Two variables:
a1x + b1y + c1 = 0 &
a2x + b2y + c2 = 0
a1 = 9, b1 = 3, c1 = 12
a2 = 18, b2 = 6, c2 = 24
Lines are coincident, Infinite Solutions
(iii) 6x – 3y + 10 = 0
2x – y + 9 = 0
Comparing the equations with standard form of Linear Equations in Two variables:
a1x + b1y + c1 = 0 &
a2x + b2y + c2 = 0
a1 = 6, b1 = -3, c1 = 10
a2 = 2, b2 = -1, c2 = 9
Lines are parallel, No Solution
Question 3
On comparing the ratios a1/a2, b1/b2 and c1/c2 find out whether the following pair of linear equations are consistent, or inconsistent.
(i) 3x + 2y = 5
2x – 3y = 7
Move the constant terms on the LHS.
3x + 2y – 5 = 0
2x – 3y – 7 = 0
Comparing the equations with standard form of Linear Equations in Two variables:
a1x + b1y + c1 = 0 &
a2x + b2y + c2 = 0
a1 = 3, b1 = 2, c1 = -5
a2 = 2, b2 = -3, c2 = -7
Consistent, Unique Solution
(ii) 2x – 3y = 8
4x – 6y = 9
Move the constant terms on the LHS.
2x – 3y – 8 = 0
4x – 6y – 9 = 0
Comparing the equations with standard form of Linear Equations in Two variables:
a1x + b1y + c1 = 0 &
a2x + b2y + c2 = 0
a1 = 2, b1 = -3, c1 = -8
a2 = 4, b2 = -6, c2 = -9
Inconsistent, No Solution
(iii) 3x/2 + 5y/3 = 7
9x – 10y = 14
Let us first remove the denominators from the equation (1)
So the equations will be:
9x + 10y – 42 = 0
9x – 10y – 14 = 0
Comparing the equations with standard form of Linear Equations in Two variables:
a1x + b1y + c1 = 0 &
a2x + b2y + c2 = 0
a1 = 9, b1 = 10, c1 = -42
a2 = 2, b2 = -1, c2 = 9
Consistent, Uunique Solution
(iv) 5x – 3y = 11
– 10x + 6y = –22
Move the constant terms on the LHS.
5x – 3y – 11 = 0
–10x + 6y + 22 = 0
Comparing the equations with standard form of Linear Equations in Two variables:
a1x + b1y + c1 = 0 &
a2x + b2y + c2 = 0
a1 = 5, b1 = -3, c1 = -11
a2 = -10, b2 = 6, c2 = 22
Consistent, Infinite Solutions
(v) 4x/3 + 2y = 8
2x + 3y = 12
Let us first remove the denominators from the equation (1)
So the equations will be:
4x + 6y – 24 = 0
2x + 3y – 12 = 0
Comparing the equations with standard form of Linear Equations in Two variables:
a1x + b1y + c1 = 0 &
a2x + b2y + c2 = 0
a1 = 4, b1 = 6, c1 = -24
a2 = 2, b2 = 3, c2 = -12
Consistent , Infinite solutions
Question 4
Which of the following pairs of linear equations are consistent/inconsistent? If consistent, obtain the solution graphically:
(i) x + y = 5, 2x + 2y = 10
Conclusion:
The equations are equivalent
Lines are coincident
System is consistent and dependent
Consistent, Infinitely Many Solutions
(ii) x – y = 8, 3x – 3y = 16
(1) and (3) have same LHS but different RHS
Conclusion
Lines are parallel
System is inconsistent
Inconsistent, No Solution
(iii) 2x + y – 6 = 0, 4x – 2y – 4 = 0
Comparing the equations with standard form of Linear Equations in Two variables:
a1x + b1y + c1 = 0 &
a2x + b2y + c2 = 0
a1 = 2, b1 = 1, c1 = -6
a2 = 4, b2 = -2, c2 = -4
Conclusion
Lines intersect each other at one point
System is consistent
Consistent, Unique Solution
(iv) 2x – 2y – 2 = 0, 4x – 4y – 5 = 0
Comparing the equations with standard form of Linear Equations in Two variables:
a1x + b1y + c1 = 0 &
a2x + b2y + c2 = 0
a1 = 2, b1 = -2, c1 = -2
a2 = 4, b2 = -4, c2 = -5
Conclusion
Lines are parallel to each other
System is inconsistent
Inconsistent, No Solution
Question 5
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.
For Garden
Let Length = x
then Width = x + 4
Half Perimeter = 36 m
Perimeter = 2 (Length + Width)
=> Length + Width = 36
Length = x = 16 m
Width = x + 4 = 20 m
Question 6
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
Two lines intersect each other if
So any of the following equations can satisfy this condition
3x + 2y – 8 = 0
3x + 3y – 8 = 0
11x + 11y + 11 = 0
You can write as many equations as you like.
(ii) Parallel lines
Two lines are parallel if
The easiest way to form the second equation is using the same equation with different constant term. For example,
2x + 3y – 1 = 0
2x + 3y – 2 = 0
2x + 3y + 3 = 0
You can write as many equations as you like.
(iii) Coincident lines
Two lines are coincident if
The easiest way to form the second equation is by multiplying the given equation with any non-zero real number. For example,
2 * ( 2x + 3y – 8 = 0 ) = 4x + 6y – 16 = 0
3 * ( 2x + 3y – 8 = 0 ) = 6x + 9y – 24 = 0
(-1) * ( 2x + 3y – 8 = 0 ) = -2x – 3y + 8 = 0
You can write as many equations as you like.
Question 7
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.
Graph
Extend the Blue and Magenta lines so that these lines intersect the x-axis. Then shade the triagular region formed between these lines and x-axis.