Class 9 – Chapter 4: Linear Equations in Two Variables (NCERT/CBSE) – Notes + Q&A + Exercise Solutions

Chapter 4: Linear Equations in Two Variables – Notes, Practice & Exercise Solutions (Class 9, NCERT/CBSE)

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1-Mark Questions (20)

1) Write the standard form of a linear equation in two variables.

\(ax+by+c=0\), where \(a,b,c\in \mathbb{R}\) and not both \(a,b\) are zero.

2) Identify \(a,b,c\) in \(2x-3y+7=0\).

\(a=2,\ b=-3,\ c=7\).

3) Is \(x+\dfrac{2}{y}=5\) a linear equation in two variables?

No, exponent of \(y\) is \(-1\) (not allowed).

4) How many solutions does \(x-2y=4\) have?

Infinitely many ordered pairs \((x,y)\).

5) Check whether \((3,2)\) satisfies \(2x+3y=12\).

\(2(3)+3(2)=12\Rightarrow\) Yes.

6) Give one solution of \(x=3y\).

\((3,1)\) (among infinitely many).

7) Write \(x=5\) in the form \(ax+by+c=0\).

\(1\cdot x+0\cdot y-5=0\).

8) State whether \((0,4)\) satisfies \(3x+2y=8\).

\(3(0)+2(4)=8\Rightarrow\) Yes.

9) Find the value of \(k\) if \((2,1)\) satisfies \(2x+3y=k\).

\(k=2\cdot2+3\cdot1=7\).

10) Write an equation whose solution set includes \((0,5)\).

Example: \(x+y=5\) (since \(0+5=5\)).

11) Is \(3x+2=0\) a linear equation in two variables?

Yes: \(3x+0\cdot y+2=0\).

12) For \(4x+3y=12\), find the intercepts.

x–intercept: \((3,0)\); y–intercept: \((0,4)\).

13) Write the ordinate of the solution \((a,b)\) of \(x+2y=7\).

Ordinate is \(b\).

14) Does \((1,-2)\) satisfy \(x-2y=5\)?

\(1-2(-2)=5\Rightarrow 1+4=5\Rightarrow\) Yes.

15) Convert \(y=\dfrac{1}{2}x-3\) to standard form.

\(x-2y-6=0\).

16) True/False: \((x,y)=(y,x)\) for all real \(x,y\).

False, unless \(x=y\).

17) Give one solution of \(5x-0\cdot y=20\).

\((4, \text{any } y)\), e.g., \((4,0)\).

18) Write the coefficient of \(y\) in \(x-3y-8=0\).

\(-3\).

19) Decide if \(x=3y+2\) is linear in two variables.

Yes; rewrite \(x-3y-2=0\).

20) State the geometric representation of a linear equation in two variables.

A straight line in the Cartesian plane.

2-Mark Questions (20)

1) Express \(x-\dfrac{y}{5}-10=0\) in the form \(ax+by+c=0\) and state \(a,b,c\).

Already in form with \(a=1,\ b=-\tfrac{1}{5},\ c=-10\). Multiplying by 5: \(5x-y-50=0\) (then \(a=5,b=-1,c=-50\)).

2) Find any four distinct solutions of \(2x+y=7\).

\((0,7),(1,5),(2,3),(3,1)\).

3) Check whether \((4,0),\ (2,0)\) satisfy \(x-2y=4\).

\((4,0): 4=4\) ✓; \((2,0): 2\ne4\) ✗.

4) Convert \(x=3y\) and \(2x=-5y\) to standard form.

\(x-3y=0\) and \(2x+5y=0\).

5) If \((k,2)\) lies on \(3x+y=11\), find \(k\).

\(3k+2=11\Rightarrow k=3\).

6) Determine the x– and y–intercepts of \(3x+2y=12\).

x–intercept \((4,0)\); y–intercept \((0,6)\).

7) Write an equation of a line with y–intercept \(5\).

Example: \(x+y=5\) (intercepts \((5,0)\) and \((0,5)\)).

8) Does \((\sqrt{2},4\sqrt{2})\) satisfy \(x-2y=4\)?

\(\sqrt{2}-2(4\sqrt{2})=\sqrt{2}-8\sqrt{2}=-7\sqrt{2}\ne4\Rightarrow\) No.

9) Write four solutions of \(\pi x+y=9\).

\((0,9),(1,9-\pi),(2,9-2\pi),(-1,9+\pi)\).

10) For \(3x+2=0\), give two solutions.

\(x=-\tfrac{2}{3}\) and \(y\) arbitrary. Examples: \(\left(-\tfrac{2}{3},0\right),\left(-\tfrac{2}{3},5\right)\).

11) Find \(c\) so that \((1,1)\) lies on \(2x-5y+c=0\).

\(2-5+c=0\Rightarrow c=3\).

12) If \(x=4y\), list three solutions with integer coordinates.

\((0,0),(4,1),(8,2)\) (many others).

13) Express \(y=\dfrac{3}{2}x-7\) in standard form and list intercepts.

\(3x-2y-14=0\). Intercepts: x–intercept \((\tfrac{14}{3},0)\); y–intercept \((0,-7)\).

14) Verify that \((0,3)\) satisfies \(x+2y=6\) and \((6,0)\) also satisfies it.

\(0+6=6\) ✓ and \(6+0=6\) ✓.

15) Find \(k\) if \((2,1)\) satisfies \(kx-4y=0\).

\(2k-4=0\Rightarrow k=2\).

16) Construct a linear equation for: “Sum of two numbers is 20”.

Let numbers be \(x,y\). Then \(x+y=20\).

17) Convert \(5=2x\) and \(y-2=0\) into \(ax+by+c=0\).

\(2x-5=0\) and \(0\cdot x+1\cdot y-2=0\).

18) Check if \((1,1)\) satisfies \(x-2y=4\).

\(1-2= -1\ne 4\Rightarrow\) No.

19) Find two different solutions of \(2x+5y=0\).

\((0,0)\) and \(\left(5,-2\right)\) (also \((-5,2)\), etc.).

20) Write an equation with x–intercept \(=6\).

Example: \(x+y=6\) (intercepts \((6,0)\) and \((0,6)\)).

3-Mark Questions (20)

1) Write a linear equation representing: “The cost of a notebook is twice the cost of a pen.” Take notebook \(=x\), pen \(=y\). Also list three solutions in integers if possible.

Equation: \(x=2y\Rightarrow x-2y=0\). Integer solutions include \((0,0),(2,1),(4,2)\).

2) Find four different solutions of \(x+2y=6\) and give the intercepts.

Solutions: \((0,3),(2,2),(4,1),(6,0)\). Intercepts: x–\((6,0)\), y–\((0,3)\).

3) If \((k, -1)\) lies on \(2x+3y=1\), find \(k\). Also verify by substitution.

\(2k+3(-1)=1\Rightarrow 2k-3=1\Rightarrow k=2\). Check: \(2(2)+3(-1)=4-3=1\) ✓.

4) The perimeter of a rectangle is 50 cm. Let length \(=x\) cm and breadth \(=y\) cm. (i) Form the linear equation. (ii) Give two possible integer solutions.

(i) \(2x+2y=50\Rightarrow x+y=25\). (ii) \((x,y)=(20,5),(18,7)\) etc.

5) Write the equation of a line whose solutions all have y–coordinate \(=4\). Give three solutions.

Equation: \(y=4\Rightarrow 0\cdot x+1\cdot y-4=0\). Solutions: \((0,4),(1,4),(-5,4)\).

6) Show that \((3,0)\) and \((0,6)\) satisfy \(2x+y=6\). Hence write two more solutions.

\(2(3)+0=6\), \(0+6=6\) ✓. More: \((1,4)\) and \((2,2)\).

7) If \(x=4y\) and \(x+y=25\), find one ordered pair \((x,y)\) satisfying both.

From \(x=4y\): \(4y+y=25\Rightarrow y=5\Rightarrow x=20\). Pair \((20,5)\).

8) For the line \(3x-2y=12\), compute: (i) x–intercept (ii) y–intercept (iii) a third point with integer coordinates on the line.

(i) \(y=0\Rightarrow x=4\Rightarrow (4,0)\). (ii) \(x=0\Rightarrow -2y=12\Rightarrow (0,-6)\). (iii) Take \(x=2\Rightarrow -2y=6\Rightarrow y=-3\Rightarrow (2,-3)\).

9) A two–digit number has tens digit \(x\) and units digit \(y\). If the number equals 7 times the sum of its digits, form the linear equation in \(x,y\). Also give one solution.

Number \(=10x+y\). Condition: \(10x+y=7(x+y)\Rightarrow 3x-6y=0\Rightarrow x=2y\). One solution in digits: \(y=3,x=6\Rightarrow 63\).

10) Find \(k\) if the x–intercept of \(kx-5y=20\) equals 4.

x–intercept at \(y=0\): \(kx=20\). Given \(x=4\Rightarrow 4k=20\Rightarrow k=5\).

11) For \(2x+5y=0\), show that \((5,-2)\) and \((-5,2)\) are solutions. Also find an intercept.

\(2(5)+5(-2)=0\) and \(2(-5)+5(2)=0\). Intercepts: both pass through origin, so only intercept is at \((0,0)\).

12) If \((a,b)\) and \((a, b+2)\) both satisfy \(x-3y=10\), show that \(b\) is unique and find it.

Substitute: \(a-3b=10\) and \(a-3(b+2)=10\Rightarrow a-3b-6=10\Rightarrow a-3b=16\). Contradiction unless \(6=0\). So both cannot satisfy unless no change in \(y\). Hence impossible except if the equation is inconsistent with the second point; so \(b\) cannot differ by 2. (Conclusion: only one of the two can lie on the line.)

13) Form a linear equation representing: “The perimeter of a square is 48 cm”. Take side \(=s\). Give two solutions in \((x,y)\) by setting \(x=s,\ y=\) perimeter.

\(4s=48\Rightarrow s=12\). As a two–variable linear relation: \(4x-y=0\) with \((x,y)=(12,48)\) and e.g. \((0,0)\) also satisfies the relation pattern.

14) Determine whether \((1,1),(2,2),(3,3)\) satisfy \(x-y=0\). What do you infer?

All three satisfy \(x=y\). Inference: infinitely many points with \(x=y\) lie on the line.

15) Find two solutions of \(x-5y=15\) where both \(x,y\) are integers.

Take \(y=0\Rightarrow x=15\Rightarrow (15,0)\). Take \(y=3\Rightarrow x=30\Rightarrow (30,3)\).

16) If \((m,2)\) and \((m,5)\) are both on \(x+ky=17\), determine \(m,k\).

Equations: \(m+2k=17\) and \(m+5k=17\Rightarrow 3k=0\Rightarrow k=0\Rightarrow m=17\).

17) A taxi charges ₹50 fixed plus ₹12 per km. Let fare \(=F\), distance \(=d\). (i) Form the equation. (ii) Find \(F\) when \(d=10\).

(i) \(F=50+12d\Rightarrow -12d+F-50=0\). (ii) \(F=50+120=170\).

18) Show that if \((x,y)\) satisfies \(x+2y=8\), then \((x+2,y+1)\) satisfies \(x+2y=10\).

Compute: \((x+2)+2(y+1)=x+2y+4=8+4=12\) (Correction: it equals 12). Hence it satisfies \(x+2y=12\), not 10.

19) If \(x-2y=4\), find \(x\) when \(y=-3\). Also find \(y\) when \(x=10\).

For \(y=-3\): \(x-2(-3)=4\Rightarrow x+6=4\Rightarrow x=-2\). For \(x=10\): \(10-2y=4\Rightarrow -2y=-6\Rightarrow y=3\).

20) The sum of two numbers is 30 and one is 6 more than twice the other. Form two equations in \(x,y\) and solve.

\(x+y=30,\ x=2y+6\Rightarrow 2y+6+y=30\Rightarrow y=8,\ x=22\). Pair \((22,8)\).

Exercise 4.1 – Solutions

1) The cost of a notebook is twice the cost of a pen. Write a linear equation in two variables. (Notebook = \(x\), Pen = \(y\))

Given \(x\) is twice \(y\): \(x=2y\ \Rightarrow\ x-2y=0\).

2) Express each in \(ax+by+c=0\) and state \(a,b,c\):
(i) \(2x+3y=9.35\) (ii) \(x-\dfrac{y}{5}-10=0\) (iii) \(-2x+3y=6\) (iv) \(x=3y\)
(v) \(2x=-5y\) (vi) \(3x+2=0\) (vii) \(y-2=0\) (viii) \(5=2x\)

(i) \(2x+3y-9.35=0:\ a=2,b=3,c=-9.35\).
(ii) \(x-\tfrac{1}{5}y-10=0:\ a=1,b=-\tfrac{1}{5},c=-10\) (or \(5x-y-50=0\)).
(iii) \(-2x+3y-6=0:\ a=-2,b=3,c=-6\).
(iv) \(x-3y=0:\ a=1,b=-3,c=0\).
(v) \(2x+5y=0:\ a=2,b=5,c=0\).
(vi) \(3x+2=0:\ a=3,b=0,c=2\).
(vii) \(y-2=0:\ a=0,b=1,c=-2\).
(viii) \(2x-5=0:\ a=2,b=0,c=-5\).

Exercise 4.2 – Solutions

1) Which option is true for \(y=3x+5\): (i) unique solution (ii) only two solutions (iii) infinitely many solutions?

(iii) Infinitely many solutions (every \(x\) gives a unique \(y\)).

2) Write four solutions each:
(i) \(2x+y=7\) (ii) \(\pi x+y=9\) (iii) \(x=4y\)

(i) \((0,7),(1,5),(2,3),(3,1)\).
(ii) \((0,9),(1,9-\pi),(2,9-2\pi),(-1,9+\pi)\).
(iii) \((0,0),(4,1),(8,2),(-4,-1)\).

3) Check which satisfy \(x-2y=4\):
(i) \((0,2)\) (ii) \((2,0)\) (iii) \((4,0)\) (iv) \((\sqrt{2},4\sqrt{2})\) (v) \((1,1)\)

(i) \(0-4=-4\ne4\) ✗ (ii) \(2-0=2\ne4\) ✗ (iii) \(4-0=4\) ✓
(iv) \(\sqrt{2}-8\sqrt{2}=-7\sqrt{2}\ne4\) ✗ (v) \(1-2=-1\ne4\) ✗

4) Find \(k\) if \((x,y)=(2,1)\) is a solution of \(2x+3y=k\).