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NCERT Exemplar · Class 9 Mathematics Linear Equations in Two Variables

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EXERCISE 4.3 1–10 (part 4 of 5)

  1. Exercise 1

    Draw the graphs of linear equations y=x\displaystyle y=x and y=−x\displaystyle y=-x on the same cartesian plane. What do you observe?

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    NCERT’s answer
    Graph of each equation is a line passing through $\displaystyle (0,0)$.
    \[y=x:\quad x=-4\Rightarrow y=-4,\quad x=0\Rightarrow y=0,\quad x=4\Rightarrow y=4 \] \[y=-x:\quad x=-4\Rightarrow y=4,\quad x=0\Rightarrow y=0,\quad x=4\Rightarrow y=-4 \] NCERT_Solution_Class9_Maths_Exemplar_Ch4_Ex4-3_Q1 Both lines pass through the origin, and each is the mirror image of the other in the \(\displaystyle x\)-axis. Answer: the two lines meet only at the origin \(\displaystyle (0,0)\), and \(\displaystyle y=x\) is perpendicular to \(\displaystyle y=-x\).

  2. Exercise 2

    Determine the point on the graph of the linear equation 2x+5y=19\displaystyle 2 x+5 y=19, whose ordinate is 112\displaystyle 1 \frac{1}{2} times its abscissa.

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    NCERT’s answer
    $\displaystyle (2,3)$
    Let the abscissa be \(\displaystyle x\), so the ordinate is \(\displaystyle y=\tfrac{3}{2}x\). \[2x+5y=19 \] \[2x+5\left(\frac{3}{2}x\right)=19 \] \[2x+\frac{15x}{2}=19 \] \[\frac{19x}{2}=19\ \Rightarrow\ x=2 \] \[y=\frac{3}{2}(2)=3 \] Answer: the point is \(\displaystyle (2,3)\).

  3. Exercise 3

    Draw the graph of the equation represented by a straight line which is parallel to the x\displaystyle x-axis and at a distance 3\displaystyle 3 units below it.

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    NCERT’s answer
    Any line parallel to $\displaystyle x$-axis and at a distance of $\displaystyle 3$ units below it is given by $\displaystyle y=-3$
    A point \(\displaystyle 3\) units below the \(\displaystyle x\)-axis has ordinate \(\displaystyle -3\), the same for every \(\displaystyle x\). \[y=-3 \] NCERT_Solution_Class9_Maths_Exemplar_Ch4_Ex4-3_Q3 Answer: \(\displaystyle y=-3\).

  4. Exercise 4

    Draw the graph of the linear equation whose solutions are represented by the points having the sum of the coordinates as 10\displaystyle 10 units.

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    NCERT’s answer
    $\displaystyle x+y=10$
    The sum of the coordinates of every solution is \(\displaystyle 10\). \[x+y=10 \] NCERT_Solution_Class9_Maths_Exemplar_Ch4_Ex4-3_Q4 Answer: \(\displaystyle x+y=10\).

  5. Exercise 5

    Write the linear equation such that each point on its graph has an ordinate 3\displaystyle 3 times its abscissa.

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    NCERT’s answer
    $\displaystyle y=3 x$
    The ordinate is \(\displaystyle 3\) times the abscissa. \[y=3x \] Answer: \(\displaystyle y=3x\).

  6. Exercise 6

    If the point (3,4)\displaystyle (3,4) lies on the graph of 3y=ax+7\displaystyle 3 y=a x+7, then find the value of a\displaystyle a.

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    NCERT’s answer
    $\displaystyle \frac{5}{3}$
    \[3y=ax+7 \] \[3(4)=a(3)+7 \] \[12=3a+7 \] \[3a=5\ \Rightarrow\ a=\frac{5}{3} \] Answer: \(\displaystyle a=\dfrac{5}{3}\).

  7. Exercise 7

    How many solution(s) of the equation 2x+1=x−3\displaystyle 2 x+1=x-3 are there on the : (i) Number line (ii) Cartesian plane

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    NCERT’s answer
    (i)
    one (ii) Infinitely many solutions
    \[2x+1=x-3 \]
    \[x=-4 \] (i) On the number line this is one point — exactly one solution.
    (ii)
    On the Cartesian plane the same equation is \(\displaystyle x+0\!\cdot\!y=-4\), true for every \(\displaystyle y\) — infinitely many solutions.
    NCERT_Solution_Class9_Maths_Exemplar_Ch4_Ex4-3_Q7
    Answer: (i) \(\displaystyle 1\) solution; (ii) infinitely many solutions.

  8. Exercise 8

    Find the solution of the linear equation x+2y=8\displaystyle x+2 y=8 which represents a point on (i) x\displaystyle x-axis (ii) y\displaystyle y-axis

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    \[x+2y=8 \]
    (i)
    On the \(\displaystyle x\)-axis, \(\displaystyle y=0\):
    \[x+2(0)=8\ \Rightarrow\ x=8 \]
    (ii)
    On the \(\displaystyle y\)-axis, \(\displaystyle x=0\):
    \[0+2y=8\ \Rightarrow\ y=4 \]
    Answer: (i) \(\displaystyle (8,0)\); (ii) \(\displaystyle (0,4)\).
    NCERT prints: (i) \(\displaystyle (4,0)\) (ii) \(\displaystyle (0,2)\) — neither satisfies \(\displaystyle x+2y=8\): \(\displaystyle 4+2(0)=4\) and \(\displaystyle 0+2(2)=4\), both short of 8.

  9. Exercise 9

    For what value of c\displaystyle c, the linear equation 2x+cy=8\displaystyle 2 x+c y=8 has equal values of x\displaystyle x and y\displaystyle y for its solution.

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    NCERT’s answer
    $\displaystyle c=\frac{8-2 x}{x}, x \neq 0$
    For a common value \(\displaystyle x = y = a \): \[2x + cy = 8 \] \[2a + ca = 8 \] \[a(2+c) = 8 \] \[a = \dfrac{8}{2+c} \] Real for every \(\displaystyle c \neq -2 \); e.g. \(\displaystyle c = 2 \) gives \(\displaystyle a = 2 \).Answer: any \(\displaystyle c \neq -2 \), with \(\displaystyle x = y = \dfrac{8}{2+c} \).

  10. Exercise 10

    Let y\displaystyle y varies directly as x\displaystyle x. If y=12\displaystyle y=12 when x=4\displaystyle x=4, then write a linear equation. What is the value of y\displaystyle y when x=5\displaystyle x=5 ?

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    NCERT’s answer
    $\displaystyle y=3 x ; \quad y=15$.
    Direct variation: \[y = kx \] \[12 = k(4) \implies k = 3 \] \[y = 3x \] \[x = 5: \quad y = 3(5) = 15 \]Answer: \(\displaystyle y = 3x \); \(\displaystyle y = 15 \) when \(\displaystyle x = 5 \).