SolveItClass 10 · NCERT

NCERT Solutions · Class 10 Mathematics Pair of Linear Equations in Two Variables

12 questions · 12 still being checked

EXERCISE 3.1 1–7 (part 1 of 3)

  1. Exercise 1

    Form the pair of linear equations in the following problems, and find their solutions graphically.
    (i)
    10\displaystyle 10 students of Class X took part in a Mathematics quiz. If the number of girls is 4\displaystyle 4 more than the number of boys, find the number of boys and girls who took part in the quiz.
    (ii)
    5\displaystyle 5 pencils and 7\displaystyle 7 pens together cost ₹ 50\displaystyle 50, whereas 7\displaystyle 7 pencils and 5\displaystyle 5 pens together cost ₹ 46. Find the cost of one pencil and that of one pen.
    NCERT’s answer
    (i)
    Required pair of linear equations is $\displaystyle x+y=10 ; x-y=4$, where $\displaystyle x$ is the number of girls and $\displaystyle y$ is the number of boys. To solve graphically draw the graphs of these equations on the same axes on graph paper. Girls =$\displaystyle 7$, Boys =3. (ii) Required pair of linear equations is $\displaystyle 5 x+7 y=50 ; 7 x+5 y=46$, where $\displaystyle x$ and $\displaystyle y$ represent the cost (in ₹) of a pencil and of a pen respectively. To solve graphically, draw the graphs of these equations on the same axes on graph paper. Cost of one pencil = ₹ $\displaystyle 3$, Cost of one pen = ₹ $\displaystyle 5$

    Working being prepared

  2. Exercise 2

    On comparing the ratios a1a2,b1b2\displaystyle \frac{a_{1}}{a_{2}}, \frac{b_{1}}{b_{2}} and c1c2\displaystyle \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\displaystyle 5 x-4 y+8=0 7x+6y−9=0\displaystyle 7 x+6 y-9=0
    (ii)
    9x+3y+12=0\displaystyle 9 x+3 y+12=0 18x+6y+24=0\displaystyle 18 x+6 y+24=0
    (iii)
    6x−3y+10=0\displaystyle 6 x-3 y+10=0 2x−y+9=0\displaystyle 2 x-y+9=0
    NCERT’s answer
    (i)
    Intersect at a point (ii) Coincident (iii) Parallel

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  3. Exercise 3

    On comparing the ratios a1a2,b1b2\displaystyle \frac{a_{1}}{a_{2}}, \frac{b_{1}}{b_{2}} and c1c2\displaystyle \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\displaystyle 3 x+2 y=5 ; \quad 2 x-3 y=7
    (ii)
    2x−3y=8;4x−6y=9\displaystyle 2 x-3 y=8 ; 4 x-6 y=9
    (iii)
    32x+53y=7;9x−10y=14\displaystyle \frac{3}{2} x+\frac{5}{3} y=7 ; 9 x-10 y=14
    (iv)
    5x−3y=11;−10x+6y=−22\displaystyle 5 x-3 y=11 ;-10 x+6 y=-22
    (v)
    43x+2y=8;2x+3y=12\displaystyle \frac{4}{3} x+2 y=8 ; 2 x+3 y=12
    NCERT’s answer
    (i)
    Consistent (ii) Inconsistent (iii) Consistent (iv) Consistent (v) Consistent

    Working being prepared

  4. Exercise 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\displaystyle x+y=5, \quad 2 x+2 y=10
    (ii)
    x−y=8,3x−3y=16\displaystyle x-y=8, \quad 3 x-3 y=16
    (iii)
    2x+y−6=0,4x−2y−4=0\displaystyle 2 x+y-6=0, \quad 4 x-2 y-4=0
    (iv)
    2x−2y−2=0,4x−4y−5=0\displaystyle 2 x-2 y-2=0,4 x-4 y-5=0
    NCERT’s answer
    (i)
    Consistent (ii) Inconsistent (iii) Consistent (iv) Inconsistent The solution of (i) above, is given by $\displaystyle y=5-x$, where $\displaystyle x$ can take any value, i.e., there are infinitely many solutions. The solution of (iii) above is $\displaystyle x=2, y=2$, i.e., unique solution.

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  5. Exercise 5

    Half the perimeter of a rectangular garden, whose length is 4\displaystyle 4 m more than its width, is 36\displaystyle 36 m. Find the dimensions of the garden.
    NCERT’s answer
    Length $\displaystyle =20 \mathrm{~m}$ and breadth $\displaystyle =16 \mathrm{~m}$.

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  6. Exercise 6

    Given the linear equation 2x+3y−8=0\displaystyle 2 x+3 y-8=0, write another linear equation in two variables such that the geometrical representation of the pair so formed is:
    (i)
    intersecting lines
    (ii)
    parallel lines
    (iii)
    coincident lines
    NCERT’s answer
    One possible answer for the three parts: (i) $\displaystyle 3 x+2 y-7=0$ (ii) $\displaystyle 2 x+3 y-12=0$ (iii) $\displaystyle 4 x+6 y-16=0$

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  7. Exercise 7

    Draw the graphs of the equations x−y+1=0\displaystyle x-y+1=0 and 3x+2y−12=0\displaystyle 3 x+2 y-12=0. Determine the coordinates of the vertices of the triangle formed by these lines and the x\displaystyle x-axis, and shade the triangular region.
    NCERT’s answer
    Vertices of the triangle are $\displaystyle (-1, 0)$, $\displaystyle (4,0)$ and $\displaystyle (2, 3)$.

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