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NCERT Exemplar · Class 9 Mathematics Linear Equations in Two Variables

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EXERCISE 4.1 1–10 (part 1 of 5)

  1. Write the correct answer in each of the following :

    Exercise 1

    The linear equation 2x−5y=7\displaystyle 2 x-5 y=7 has (A) A unique solution (B) Two solutions (C) Infinitely many solutions (D) No solution

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    NCERT’s answer
    (C)
    (C) Infinitely many solutions. \[y = \frac{2x-7}{5} \] Every real \(\displaystyle x\) yields a real \(\displaystyle y\), so the line \(\displaystyle 2x-5y=7\) has infinitely many points satisfying it.

  2. Exercise 2

    The equation 2x+5y=7\displaystyle 2 x+5 y=7 has a unique solution, if x,y\displaystyle x, y are : (A) Natural numbers (B) Positive real numbers (C) Real numbers (D) Rational numbers

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    NCERT’s answer
    (A)
    (A) Natural numbers. \[2x+5y=7,\quad x,y\in\mathbb{N} \implies x=1,\ y=1 \] No other natural pair fits; over positive reals or all reals the equation has infinitely many solutions.

  3. Exercise 3

    If (2,0)\displaystyle (2,0) is a solution of the linear equation 2x+3y=k\displaystyle 2 x+3 y=k, then the value of k\displaystyle k is (A) 4\displaystyle 4 (B) 6\displaystyle 6 (C) 5\displaystyle 5 (D) 2\displaystyle 2

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    NCERT’s answer
    (A)
    (A) \(\displaystyle 4\). \[2(2)+3(0)=k \implies k=4 \]

  4. Exercise 4

    Any solution of the linear equation 2x+0y+9=0\displaystyle 2 x+0 y+9=0 in two variables is of the form (A) (−92,m)\displaystyle \left(-\frac{9}{2}, m\right) (B) (n,−92)\displaystyle \left(n,-\frac{9}{2}\right) (C) (0,−92)\displaystyle \left(0,-\frac{9}{2}\right) (D) (−9,0)\displaystyle (- 9, 0)

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    NCERT’s answer
    (A)
    (A) \(\displaystyle \left(-\dfrac{9}{2},\,m\right)\). \[2x+9=0 \implies x=-\frac{9}{2} \] the coefficient of \(\displaystyle y\) is \(\displaystyle 0\), so \(\displaystyle y\) is free and can be any value \(\displaystyle m\).

  5. Exercise 5

    The graph of the linear equation 2x+3y=6\displaystyle 2 x+3 y=6 cuts the y\displaystyle y-axis at the point (A) (2,0)\displaystyle (2,0) (B) (0,3)\displaystyle (0,3) (C) (3,0)\displaystyle (3,0) (D) (0,2)\displaystyle (0,2)

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    NCERT’s answer
    (D)
    (D) \(\displaystyle (0,2)\). \[x=0 \implies 3y=6 \implies y=2 \]

  6. Exercise 6

    The equation x=7\displaystyle x=7, in two variables, can be written as (A) 1.x+1.y=7\displaystyle 1 . x+1 . y=7 (B) 1. x+0.y=7\displaystyle x+0 . y=7 (C) 0.x+1.y=7\displaystyle 0 . x+1 . y=7 (D) 0.x+0.y=7\displaystyle 0 . x+0 . y=7

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    NCERT’s answer
    (B)
    (B) \(\displaystyle 1.x+0.y=7\). \[1\cdot x + 0\cdot y = x \] equals \(\displaystyle x=7\) for every value of \(\displaystyle y\), matching the given equation exactly.

  7. Exercise 7

    Any point on the x\displaystyle x-axis is of the form (A) (x,y)\displaystyle (x, y) (B) (0,y)\displaystyle (0, y) (C) (x,0)\displaystyle (x, 0) (D) (x,x)\displaystyle (x, x)

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    NCERT’s answer
    (C)
    (C) \(\displaystyle (x,0)\). Every point on the \(\displaystyle x\)-axis has ordinate \(\displaystyle 0\), with the abscissa \(\displaystyle x\) free to take any value.

  8. Exercise 8

    Any point on the line y=x\displaystyle y=x is of the form (A) (a,a)\displaystyle (a, a) (B) (0,a)\displaystyle (0, a) (C) (a,0)\displaystyle (a, 0) (D) (a,−a)\displaystyle (a,-a)

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    NCERT’s answer
    (A)
    (A) \(\displaystyle (a,a)\). \[y=x \] forces the two coordinates to be equal, so a point on this line takes the form \(\displaystyle (a,a)\).

  9. Exercise 9

    The equation of x\displaystyle x-axis is of the form (A) x=0\displaystyle x=0 (B) y=0\displaystyle y=0 (C) x+y=0\displaystyle x+y=0 (D) x=y\displaystyle x=y

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    NCERT’s answer
    (B)
    (B) \(\displaystyle y=0\). Every point on the \(\displaystyle x\)-axis has ordinate \(\displaystyle 0\), and \(\displaystyle y=0\) is exactly the equation satisfied by all such points.

  10. Exercise 10

    The graph of y=6\displaystyle y=6 is a line (A) parallel to x\displaystyle x-axis at a distance 6\displaystyle 6 units from the origin (B) parallel to y\displaystyle y-axis at a distance 6\displaystyle 6 units from the origin (C) making an intercept 6\displaystyle 6 on the x\displaystyle x-axis. (D) making an intercept 6\displaystyle 6 on both the axes.

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    NCERT’s answer
    (A)
    (A) parallel to \(\displaystyle x\)-axis at a distance \(\displaystyle 6\) units from the origin. \[y=6 \] holds for every \(\displaystyle x\), giving a horizontal line \(\displaystyle 6\) units above the \(\displaystyle x\)-axis.