LINEAR EQUATIONS IN TWO VARIABLES I

LINEAR EQUATIONS IN TWO VARIABLES I

2.1 LINEAR EQUATIONS IN TWO VARIABLES :

An equation of the form $Ax+By+C=0$ is called a linear equation.

Where A is called coefficient of $x, B$ is called coefficient of $y$ and $C$ is the constant term (free form $x & y$ )

A, B, C, $\in R[\in \to$ belongs, to $R \to$ Real No. $]$

But $A$ and $B$ ca not be simultaneously zero.

If $A \neq 0, B=0$ equation will be of the form $A x+C=0$. [Line || to Y-axis]

If $A=0, B \neq 0$, equation will be of the form $By+C=0$.

If $A \neq 0, B \neq 0, C=0$ equation will be of the form $A x+B y=0$. [Line || to X-axis]

If $A \neq 0, B \neq C, C \neq 0$ equation will be of the form $A x+B y+C=0$. [Line passing through origin]

It is called a linear equation in two variable because the two unknown ( $x & y$ ) occurs only in the first power, and the product of two unknown equalities does not occur.

Since it involves two variable therefore a single equation will have infinite set of solution i.e. indeterminate solution. So we require a pair of equation i.e. simultaneous equations.

LINEAR EQUATIONS IN TWO VARIABLES I

Standard form of linear equation : (Standard form refers to all positive coefficient)

$a_1 x+b_1 y+c_1=0$ …(i)

$a_2 x+b_2 y+c_2=0$ …(ii)

For solving such equations we have three methods.

(i) Elimination by substitution $\quad$

(ii) Elimination by equating the coefficients

(iii) Elimination by cross multiplication.

LINEAR EQUATIONS IN TWO VARIABLES I

2.1 Elimination By Substitution :

Ex. 1 Solve $x+4 y=14 \ldots . .(i)$

$ 7 x-3 y=5 \ldots .\text { (ii) } $

LINEAR EQUATIONS IN TWO VARIABLES I

Sol. From equation (i) $x=14-4 y$ …(iii)

Substitute the value of $x$ in equation (ii)

$ \begin{matrix} \Rightarrow & 7(14-4 y)-3 y=5 & \Rightarrow & 98-28 y-3 y=5 \\ \Rightarrow & 98-31 y=5 \quad \Rightarrow \quad 93=31 y & \Rightarrow & y=\frac{93}{31} \Rightarrow y=3 \end{matrix} $

Now substitute value of $y$ in equation (iii)

$ \begin{matrix} \Rightarrow & 7 x-3(3)=5 & \Rightarrow & 7 x-3(3)=5 \\ \Rightarrow & 7 x=14 & \Rightarrow & x=\frac{14}{7}=2 \quad \text { So, solution is } x=2 \text { and } y=3 \end{matrix} $

LINEAR EQUATIONS IN TWO VARIABLES I

2.1 (b) Elimination by Equating the Coefficients :

Ex. 2 Solve $9 x-4 y=8 \ldots .$. (i)

$ 13 x+7 y=101 \ldots \text { (ii) } $

LINEAR EQUATIONS IN TWO VARIABLES I

Sol. Multiply equation (i) by 7 and equation (ii) by 4 , we get

$ \begin{matrix} \text { Add } & 63 x-28 y & =56\\ & 52 x+28 y & =404 \end{matrix} $

$ 115 x \quad=460 $ $ \Rightarrow \quad x=\frac{460}{115} \Rightarrow x=4 $

Substitute $x=4$ in equation (i)

$ 9(4)-4 y=8 \quad \Rightarrow \quad 36-8=4 y \quad \Rightarrow \quad 28=4 y \Rightarrow \quad y=\frac{28}{4}=7 $

So, solution is $x=4$ and $y=7$.

LINEAR EQUATIONS IN TWO VARIABLES I

2.1 (c) Elimination by Cross Multiplication :

$a_1 x+b_1 y+c_1=0$

$a_2 x+b_2 y+c_2=0$ $ \quad [\because \frac{a_1}{a_2} \neq \frac{b_1}{b_2}] $

$\frac{x}{b_1 c_2-b_2 c_1}=\frac{y}{a_2 c_1-a_1 c_2}=\frac{1}{a_1 b_2-a_2 b_1} \Rightarrow \therefore \frac{x}{b_1 c_2-b_2 c_1}=\frac{1}{a_1 b_2-a_2 b_1}$

$\Rightarrow \quad x=\frac{b_1 c_2-b_2 c_1}{a_1 b_2-a_2 b_1}$

Also, $\frac{y}{a_2 c_1-a_1 c_2}=\frac{1}{a_1 b_2-a_2 b_1} \quad \therefore \quad y=\frac{a_2 c_1-a_1 c_2}{a_1 b_2-a_2 b_1}$

LINEAR EQUATIONS IN TWO VARIABLES I

Ex 3. Solve $ 3x+2y+25=0$ …(i)

$x+y+15=0$ …(ii)

LINEAR EQUATIONS IN TWO VARIABLES I

Sol. Here, $a_1=3 \ b_1=2, \ c_1=25$

$ a_2=1 \ b_2=1,\ c_2=15 $

$ \frac{x}{2 \times 15-25 \times 1}=\frac{y}{25 \times 1-15 \times 3}=\frac{1}{3 \times 1-2 \times 1} ; \frac{x}{30-25}=\frac{y}{25-45}=\frac{1}{3-2} $

$\frac{x}{5}=\frac{y}{-20}=\frac{1}{1}$ …(i)

$ \frac{x}{5}=1, \frac{y}{-20}=\frac{1}{1} $

$ X=5, y=-20 \quad \text { So, solution is } x=5 \text { and } y=-20 \text {. } $

LINEAR EQUATIONS IN TWO VARIABLES I

2.2 CONDITIONS FOR SOLVABILITY (OR CONSISTENCY) OF SYSTEM OF EQUATIONS:

2.2 (a) Unique Solution :

Two lines $a_1+b_1 y+c_1=0$ and $a_2 x+b_2 y+c_2=0$, if the denominator $a_1 b_2-a_2 b_1 \neq 0$ then the given system of equations have unique solution (i.e. only one solution) and solutions are said to be consistent.

$\therefore \quad a_1 b_2-a_2 b_1 \neq 0 \quad \Rightarrow \quad \frac{a_1}{b_2} \neq \frac{b_1}{b_2}$

LINEAR EQUATIONS IN TWO VARIABLES I

2.2 (b) No Solution :

Two lines $a_1 x+b_1 y+c_1=0$ and $a_2 x+b_2 y+c_2=0$, if the denominator $a_1 b_2-a_2 b_1=0$ then the given system of equations have no solution and solutions are said to be consistent.

$\therefore \quad a_1 b_2-a_2 b_1 \neq 0 \Rightarrow \quad \frac{a_1}{a_2} \neq \frac{b_1}{b_2}$

LINEAR EQUATIONS IN TWO VARIABLES I

2.2 (c) Many Solution (Infinite Solutions)

Two lines $a_1 x+b_1 y+c_1=0$ and $a_2 x+b_2 y+c_2=0$, if $\frac{a_1}{a_2}=\frac{b_1}{b_2}=-$ then system of equations has many solution and solutions are said to be consistent.

LINEAR EQUATIONS IN TWO VARIABLES I

Ex. 4 Find the value of ’ $P$ ’ for which the given system of equations has only one solution (i.e. unique solution). $P x-y=2$ $6 x-2 y=3$

LINEAR EQUATIONS IN TWO VARIABLES I

Sol. $\quad a_1=P,\ b_1=-1,\ c_1=-2$

$a_2=6\ b_2=-2,\ c_2=-3$

Conditions for unique solution is $\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$

$\Rightarrow \quad \frac{P}{6} \neq \frac{+1}{+2} \quad \Rightarrow \quad P \neq \frac{6}{2} \Rightarrow \quad P \neq 3 \quad \therefore P$ can have all real values except 3 .

LINEAR EQUATIONS IN TWO VARIABLES I

Ex. 5 Find the value of $k$ for which the system of linear equation

$kx+4 y=k-4$

$16 x+k y=k$ has infinite solution.

LINEAR EQUATIONS IN TWO VARIABLES I

Sol. $\quad a_1=k,\ b_1=4,\ c_1=-(k-4)$

$a_2=16,\ b_2=k,\ c_2=-k$

Here condition is $\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$

$ \begin{aligned} & \Rightarrow \frac{k}{16}=\frac{4}{k}=\frac{(k-4)}{(k)} \quad \Rightarrow \quad \frac{k}{16}=\frac{4}{k} \text { also } \quad \frac{4}{k}=\frac{k-4}{k} \\ & \Rightarrow \quad k^{2}=64 \quad \Rightarrow \quad 4 k=k^{2}-4 k \\ & \Rightarrow k= \pm 8 \quad \Rightarrow \quad k(k-8)=0 \end{aligned} $

$k=0$ or $k=8$ but $k=0$ is not possible other wise equation will be one variable.

$\therefore \quad k=8$ is correct value for infinite solution.

LINEAR EQUATIONS IN TWO VARIABLES I

Ex. 6 Determine the value of $k$ so that the following linear equations has no solution.

$ \begin{aligned} & (3 x+1) x+3 y-2=0 \\ & (k^{2}+1) x+(k-2) y-5=0 \end{aligned} $

LINEAR EQUATIONS IN TWO VARIABLES I

Sol. Here $a_1=3 k+1, b_1=3$ and $c_1=-2$

$ a_2=k^{2}+1, b_2=k-2 \text { and } c_2=-5 $ For no solution, condition is $\frac{a_1}{a_2}=\frac{b_1}{b_2} \neq \frac{c_1}{c_2}$

$\frac{3 k+1}{k^{2}+1}=\frac{3}{k-2} \neq \frac{-2}{-5}$ $\Rightarrow \quad \frac{3 k+1}{k^{2}+1}=\frac{3}{k-2}$ and $\frac{3}{k-2} \neq \frac{2}{5}$

Now, $\quad \frac{3 k+1}{k^{2}+1}=\frac{3}{k-2}$

$\Rightarrow(3 k+1)(k-2)=3(k^{2}+1)$ $\Rightarrow \quad 3 k^{2}-5 k-2=3 k^{2}+3$

$\Rightarrow-5 k-2=3$ $\Rightarrow \quad-5 k=5$

$\Rightarrow k=-1$ Clearly, $\frac{3}{k-2} \neq \frac{2}{5}$ for $k=-1$.

Hence, the given system of equations will have no solution for $k=-1$.

LINEAR EQUATIONS IN TWO VARIABLES I

DAILY PRACTIVE PROVBLEMS 2

OBJECTIVE DPP - 2.1

1. The equations $3 x-5 y+2=0$, and $6 x+4=10 y$ have :

(A) No solution $\quad$

(B) A single solution

(C) Two solutions $\quad$

(D) An infinite number of solution

LINEAR EQUATIONS IN TWO VARIABLES I

LINEAR EQUATIONS IN TWO VARIABLES I

2. If $p+q=1$ and the ordered pair (p, q) satisfy $3 x+2 y=1$ then is also satisfies :

(A) $3 x+4 y=5$ $\quad$

(B) $5 x+4 y=4$

(C) $5 x+5 y=4$ $\quad$

(D) None of these.

LINEAR EQUATIONS IN TWO VARIABLES I

LINEAR EQUATIONS IN TWO VARIABLES I

3. If $x=y, 3 x-y=4$ and $x+y+x=6$ then the value of $z$ is :

(A) 1 $\quad$

(B) 2

(C) 3 $\quad$

(D) 4

LINEAR EQUATIONS IN TWO VARIABLES I

LINEAR EQUATIONS IN TWO VARIABLES I

4. The system of linear equation $a x+b y=0, c x+d y=0$ has no solution if :

(A) ad - bc $>0$ $\quad$

(B) ad - bc $<0$

(C) $a d+b c=0$ $\quad$

(D) $ad-bc=0$

LINEAR EQUATIONS IN TWO VARIABLES I

LINEAR EQUATIONS IN TWO VARIABLES I

5. The value of $k$ for which the system $k x+3 y=7$ and $2 x-5 y=3$ has no solution is :

(A) $7 $& $k=-\frac{3}{14}$ $\quad$

(B) $4$ & $ k=\frac{3}{14}$

(C) $\frac{6}{5}$ & $k \neq \frac{14}{3}$ $\quad$

(D) $-\frac{6}{5}$ & $k \neq \frac{14}{3}$

LINEAR EQUATIONS IN TWO VARIABLES I

LINEAR EQUATIONS IN TWO VARIABLES I

6. If $29 x+37 y=103,37 x+29 y=95$ then:

(A) $x=1, y=2$ $\quad$

(B) $x=2, y=1$

(C) $x=2, y=3$ $\quad$

(D) $x=3, y=2$

LINEAR EQUATIONS IN TWO VARIABLES I

LINEAR EQUATIONS IN TWO VARIABLES I

7. On solving $\frac{25}{x+y}-\frac{3}{x-y}=1, \frac{40}{x+y}+\frac{2}{x-y}=5$ we get :

(A) $x=8, y=6$ $\quad$

(B) $x=4, y=6$

(C) $x=6, y=4$ $\quad$

(D) None of these

LINEAR EQUATIONS IN TWO VARIABLES I

LINEAR EQUATIONS IN TWO VARIABLES I

8. If the system $2 x+3 y-5=0,4 x+k y-10=0$ has an infinite number of solutions then :

(A) $k=\frac{3}{2}$ $\quad$

(B) $k \neq \frac{3}{2}$

(C) $k \neq 6$ $\quad$

(D) $k=6$

LINEAR EQUATIONS IN TWO VARIABLES I

LINEAR EQUATIONS IN TWO VARIABLES I

9. The equation $x+2 y=4$ and $2 x+y=5$

(A) Are consistent and have a unique solution $\quad$

(B) Are consistent and have infinitely many solution

(C) are inconsistent $\quad$

(D) Are homogeneous linear equations

LINEAR EQUATIONS IN TWO VARIABLES I

LINEAR EQUATIONS IN TWO VARIABLES I

10. If $\frac{1}{x}-\frac{1}{y}=\frac{1}{z}$ then $z$ will be :

(A) $y-x$ $\quad$

(B) $x-y$

(C) $\frac{y-x}{x y}$ $\quad$

(D) $\frac{x y}{y-x}$

LINEAR EQUATIONS IN TWO VARIABLES I

LINEAR EQUATIONS IN TWO VARIABLES I

SUBJECTIVE DPP 2.2

Solve each of the following pair of simultaneous equations.

1. $\frac{x}{3}+\frac{y}{12}=\frac{7}{2}$ and $\frac{x}{6}-\frac{y}{8}=\frac{6}{8}$

LINEAR EQUATIONS IN TWO VARIABLES I

Sol. 1. $x=9, y=6$

LINEAR EQUATIONS IN TWO VARIABLES I

2. $0.2 x+0.3 y=0.11=0$, $0.7 x-0.5 y+0.08=0$

LINEAR EQUATIONS IN TWO VARIABLES I

Sol. 2. $x=0.1,y=0.3 $

LINEAR EQUATIONS IN TWO VARIABLES I

3. $ 3 \sqrt{2} x-5 \sqrt{3} y+\sqrt{5}=0 ; \quad 2 \sqrt{3} x+7 \sqrt{2} y-2 \sqrt{5}=0$

LINEAR EQUATIONS IN TWO VARIABLES I

Sol. 3. $x=\frac{10 \sqrt{5}-7 \sqrt{10}}{72} y=\frac{2 \sqrt{15}+6 \sqrt{10}}{72}$

LINEAR EQUATIONS IN TWO VARIABLES I

4. $ \frac{x}{3}+y=1.7 \quad$ and $\frac{11}{x+\frac{y}{3}}=10 \forall[x+\frac{y}{3} \neq 0]$

LINEAR EQUATIONS IN TWO VARIABLES I

Sol. 4. $ x=0.6, y=1.5$

LINEAR EQUATIONS IN TWO VARIABLES I

5. Prove that the positive square root of the reciprocal of the solutions of the equations $\frac{3}{x}+\frac{5}{y}=29$ and $\frac{7}{x}-\frac{4}{y}=5(x \neq 0, y \neq 0)$ satisfy both the equation $2(\sqrt{3} x+4)-3(4 y-5)=5$ & $7(\frac{9 x}{\sqrt{3}}+8)+5(7 y-25)=64$

LINEAR EQUATIONS IN TWO VARIABLES I

6. For what value of $a$ and $b$, the following system of equations have an infinite no. of solutions. $2 x+3 y=7$; $(a-b) x+(a+b)+b-2$

LINEAR EQUATIONS IN TWO VARIABLES I

Sol. 6. $ a=5, b=1$

LINEAR EQUATIONS IN TWO VARIABLES I

7. Solve : (i) $\frac{7}{x^{3}}-\frac{6}{2^{y}}=15 ; \frac{8}{3^{x}}=\frac{9}{2^{y}}$ $\quad$ (ii) $119 x-381 y=643 ; 381 x-119 y=-143$

LINEAR EQUATIONS IN TWO VARIABLES I

Sol. 7. $ (i) x=-2, y=-3 (ii) x=-1, y=-2 $

LINEAR EQUATIONS IN TWO VARIABLES I

8. Solve: $\frac{b x}{a}-\frac{a y}{b}+a+b=0 ; b x-a y+2 a b=0$

LINEAR EQUATIONS IN TWO VARIABLES I

Sol. 8. $ x = - a, y = b $

LINEAR EQUATIONS IN TWO VARIABLES I

9. Solve : $\frac{1}{3 x}+\frac{1}{5 y}=1 ; \frac{1}{5 x}+\frac{1}{3 y}=1 \frac{2}{15}$

LINEAR EQUATIONS IN TWO VARIABLES I

Sol. 9. $x=\frac{2}{3}, y=\frac{2}{5}$

LINEAR EQUATIONS IN TWO VARIABLES I

10. Solve $x-y+z=6$

$ x-22 y-2 z=5 $

$ 2 x+y-3 z=1 $

LINEAR EQUATIONS IN TWO VARIABLES I

Sol. 10. $x = 3, y = - 2 , x = 1$

LINEAR EQUATIONS IN TWO VARIABLES I

11. Solve, $p x+q y=r$ and $q x=1+r$

LINEAR EQUATIONS IN TWO VARIABLES I

Sol. 11. $x=\frac{q+r(p+q)}{p^{2}+q^{2}}, y=\frac{r(q-p)-p}{p^{2}+q^{2}}$

LINEAR EQUATIONS IN TWO VARIABLES I

12. Find the value of $k$ for which the given system of equations

(A) has a Unique solution. $\quad$ (B) becomes consistent.

(i) $3 x+5 y=12$ $\quad$ (ii) $3 x-7 y=6$

$4 x-7 y=k$ $\quad$ $21 x-49 y=1-1$

LINEAR EQUATIONS IN TWO VARIABLES I

Sol. 12. (a) k is any real number (b) k = $41$

LINEAR EQUATIONS IN TWO VARIABLES I

13. Find the value of $k$ for which the following system of linear equation becomes infinitely many solution. or represent the coincident lines.

(i) $6 x+3 y=k-3$ $\quad$(ii) $x+2 y+7=0$

$2 k x+6 y=6$ $\quad$ $2 x+ky+14=0$

LINEAR EQUATIONS IN TWO VARIABLES I

Sol. 13. (a) $k=6(b) k=4$

LINEAR EQUATIONS IN TWO VARIABLES I

14. Find the value of $k$ or $C$ for which the following systems of equations be in consistent or no solution.

(i) $2 x ky+k+2=0$ $\quad$ (ii) $C x+3 y=3$

$kx+8 y+3 k=0$ $\quad$ $12 x+C y=6$

LINEAR EQUATIONS IN TWO VARIABLES I

Sol. 14. (a) k = $- 4 (b) C = - 6 $

LINEAR EQUATIONS IN TWO VARIABLES I

15. Solve for $x$ and $y$ : $(a-b) x+(a+b) y=a^{2}-2 a b-b^{2}$

$(a+b)(x+y)=a^{2}+b^{2}$

[CBSE - 2008]

LINEAR EQUATIONS IN TWO VARIABLES I

Sol. 15. $x=a+b, y=-\frac{2 ab}{a+b}$

LINEAR EQUATIONS IN TWO VARIABLES I

16. Solve for $x$ and $y$ :

$37 x+43 y=123$

$43 x+37 y=117$ $\quad$

[CBSE - 2008]

LINEAR EQUATIONS IN TWO VARIABLES I

Sol. 16. $x=1, y=2$