Chapter 3 · Pair of Linear Equations

NCERT Class 10 Maths Exercise 3.1

All 7 questions of NCERT Class 10 Maths Exercise 3.1 (Pair of Linear Equations), in the book's order, each with its answer and a step-by-step solution.

1. Q1
   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.
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   There are $3$ boys and $7$ girls (the lines $x+y=10$ and $y-x=4$ meet at $(3,7)$).

   Step-by-step solution →
2. Q2
   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:

   1. (i)$5x-4y+8=0$ ​and $7x+6y-9=0$
   2. (ii)$9x+3y+12=0$ and $18x+6y+24=0$
   3. (iii)$6x-3y+10=0$ and $2x-y+9=0$
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   (i) The lines intersect at a point; (ii) The lines are coincident; (iii) The lines are parallel.

   Step-by-step solution →
3. Q3
   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 pairs of linear equations are consistent or ​inconsistent:

   1. (i)$3x+2y=5$ ​and $2x-3y=7$
   2. (ii)$2x-3y=8$ and $4x-6y=9$ ​
   3. (iii)$\frac{3}{2}x+\frac{5}{3}y=7$ and $9x-10y=14$
   4. (iv)$5x-3y=11$ and $-10x+6y=-22$ ​
   5. (v)$\frac{4}{3}x+2y=8$ and $2x+3y=12$
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   (i) Consistent; (ii) Inconsistent; (iii) Consistent; (iv) Consistent (infinitely many solutions); (v) Consistent (infinitely many solutions).

   Step-by-step solution →
4. Q4
   Which of the ​following pairs ​of ​linear equations are consistent/inconsistent? ​If consistent, ​obtain ​the solution graphically:

   1. (i)$x+y=5$ and $2x+2y=10$
   2. (ii)$x-y=8$ ​and $3x-3y=16$
   3. (iii)$2x+y-6=0$ and $4x-2y-4=0$
   4. (iv)$2x-2y-2=0$ and $4x-4y-5=0$
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   (i) Consistent — coincident lines, infinitely many solutions; (ii) Inconsistent — no solution; (iii) Consistent — unique solution $(2,2)$; (iv) Inconsistent — no solution.

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5. 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.
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   The garden is $20$ m long and $16$ m wide.

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6. Q6
   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:

   1. (i)intersecting lines
   2. (ii)parallel lines
   3. (iii)coincident lines
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   (i) Intersecting: $3x+2y-7=0$; (ii) Parallel: $4x+6y-5=0$; (iii) Coincident: $4x+6y-16=0$. (Any equation meeting the same ratio condition is acceptable.)

   Step-by-step solution →
7. 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.
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   The vertices of the triangle are $A(-1,0)$, $B(2,3)$ and $C(4,0)$; the enclosed region is shaded in the graph.

   Step-by-step solution →