Padhona

Mathematics · Pair of Linear Equations in Two Variables · NCERT Exercises

Exercise 3.1

Complete, independently verified solutions for NCERT Exercise 3.1.

Mathematics · Chapter 3 · NCERT Exercises

Download PDF
Verified NCERT questions and solutionsSolutions are checked independently and follow the supplied NCERT chapter edition.

Jump navigator

Choose a question

Question

Question 1

graphical method · word problems

Form equations and solve graphically.

  1. Ten students joined a quiz. Girls were 4 more than boys.

  2. Five pencils and seven pens cost ₹50; seven pencils and five pens cost ₹46.

Solution

  1. x+y=10, y−x=4x+y=10,\ y-x=4

    Plot (0,10),(10,0) and (0,4),(3,7).

    QuizThe intersection gives 3 boys and 7 girls.y − x = 4(3, 7)
    The intersection gives 3 boys and 7 girls.

    The intersection is (3,7).

  2. 5x+7y=50, 7x+5y=465x+7y=50,\ 7x+5y=46

    Plot (3,5),(10,0) and (3,5),(−2,12).

    CostsThe intersection gives ₹3 per pencil and ₹5 per pen.(3, 5)
    The intersection gives ₹3 per pencil and ₹5 per pen.

    The intersection is (3,5); both costs verify.

Final answer

There were 3 boys and 7 girls. A pencil costs ₹3 and a pen ₹5.

Question

Question 2

coefficient ratios · line relationships

Use coefficient ratios to classify each pair.

  1. 5x−4y+8=0; 7x+6y−9=05x-4y+8=0;\ 7x+6y-9=0
  2. 9x+3y+12=0; 18x+6y+24=09x+3y+12=0;\ 18x+6y+24=0
  3. 6x−3y+10=0; 2x−y+9=06x-3y+10=0;\ 2x-y+9=0

Solution

  1. 57≠−46\frac57\ne\frac{-4}{6}

    Intersecting.

  2. 918=36=1224\frac9{18}=\frac36=\frac{12}{24}

    Coincident.

  3. 62=−3−1≠109\frac62=\frac{-3}{-1}\ne\frac{10}{9}

    Parallel.

Question

Question 3

consistent equations · coefficient ratios

Use coefficient ratios to decide consistency.

  1. 3x+2y=5; 2x−3y=73x+2y=5;\ 2x-3y=7
  2. 2x−3y=8; 4x−6y=92x-3y=8;\ 4x-6y=9
  3. 32x+53y=7; 9x−10y=14\frac32x+\frac53y=7;\ 9x-10y=14
  4. 5x−3y=11; −10x+6y=−225x-3y=11;\ -10x+6y=-22
  5. 43x+2y=8; 2x+3y=12\frac43x+2y=8;\ 2x+3y=12

Solution

  1. Coefficient ratios differ: consistent, unique solution.

  2. x and y ratios agree but constant ratio differs: inconsistent.

  3. Coefficient ratios differ: consistent, unique solution.

  4. All ratios agree: dependent and consistent, infinitely many solutions.

  5. All ratios are 2/3: dependent and consistent, infinitely many solutions.

Question

Question 4

graphical method · consistency

Classify each pair; obtain graphical solutions for consistent pairs.

  1. x+y=5; 2x+2y=10x+y=5;\ 2x+2y=10
  2. x−y=8; 3x−3y=16x-y=8;\ 3x-3y=16
  3. 2x+y−6=0; 4x−2y−4=02x+y-6=0;\ 4x-2y-4=0
  4. 2x−2y−2=0; 4x−4y−5=02x-2y-2=0;\ 4x-4y-5=0

Solution

  1. Coincident pairThe coincident pair has infinitely many solutions.
    The coincident pair has infinitely many solutions.

    Coincident: consistent with infinitely many solutions.

  2. Parallel: inconsistent, no solution.

  3. Unique pairThe lines intersect at (2, 2).4x − 2y = 4(2, 2)
    The lines intersect at (2, 2).

    Intersecting: consistent with unique solution (2,2), which verifies both equations.

  4. Parallel: inconsistent, no solution.

Question

Question 5

word problem

Half the perimeter of a rectangular garden is 36 m and its length is 4 m more than its width. Find its dimensions.

Solution

Let length be l and width w.

l+w=36,l−w=4l+w=36,\qquad l-w=4
2l=40⇒l=20,w=162l=40\Rightarrow l=20,\qquad w=16

The sum is 36 and difference is 4.

Final answer

Length 20 m; width 16 m.

Question

Question 6

forming equations · line relationships

With 2x + 3y − 8 = 0, give another equation producing each line relationship.

  1. Intersecting

  2. Parallel

  3. Coincident

Solution

One valid set is:

  1. x+y−1=0x+y-1=0

    Coefficient ratios differ.

  2. 4x+6y−9=04x+6y-9=0

    x and y ratios agree; constant ratio differs.

  3. 4x+6y−16=04x+6y-16=0

    All coefficients are doubled.

Question

Question 7

graphical method · coordinate geometry

Draw x − y + 1 = 0 and 3x + 2y − 12 = 0. Find the triangle vertices formed with the x-axis and shade the region.

Solution

Use (−1,0),(0,1),(2,3) for the first line and (4,0),(0,6),(2,3) for the second.

TriangleThe triangle vertices are (−1, 0), (4, 0), and (2, 3).x − y + 1 = 03x + 2y − 12 = 0(−1, 0)(4, 0)(2, 3)
The triangle vertices are (−1, 0), (4, 0), and (2, 3).

The x-intercepts and common point give the three vertices.

Final answer

Vertices: (−1,0), (4,0), (2,3).