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 )

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

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

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

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

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

If $A\ne 0,B\ne C,C\ne 0$ equation will be of the form $Ax+By+C=0$. [Line passing through origin]

It is called a linear equation in two variable because the two unknown ( ) 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 $$

(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 $x+4y=14\dots ..(i)$

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

LINEAR EQUATIONS IN TWO VARIABLES I

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

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

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

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

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

LINEAR EQUATIONS IN TWO VARIABLES I

2.1 (b) Elimination by Equating the Coefficients :

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

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

LINEAR EQUATIONS IN TWO VARIABLES I

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

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

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

Substitute $x=4$ in equation (i)

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

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

LINEAR EQUATIONS IN TWO VARIABLES I

2.1 (c) Elimination by Cross Multiplication :

$a_{1}x+b_{1}y+c_1=0$

$a_{2}x+b_{2}y+c_2=0$ $[\because \frac{a_1}{a_2}\ne \frac{b_1}{b_2}]$

$\frac{x}{b_1{c}_2-b_2{c}_1}=\frac{y}{a_2{c}_1-a_1{c}_2}=\frac{1}{a_1{b}_2-a_2{b}_1}\Rightarrow \therefore \frac{x}{b_1{c}_2-b_2{c}_1}=\frac{1}{a_1{b}_2-a_2{b}_1}$

$\Rightarrow x=\frac{b_1{c}_2-b_2{c}_1}{a_1{b}_2-a_2{b}_1}$

Also, $\frac{y}{a_2{c}_1-a_1{c}_2}=\frac{1}{a_1{b}_2-a_2{b}_1}\therefore y=\frac{a_2{c}_1-a_1{c}_2}{a_1{b}_2-a_2{b}_1}$

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, $a_1=3b_1=2,c_1=25$

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

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

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

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

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

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 $a_1{b}_2-a_2{b}_1\ne 0$ then the given system of equations have unique solution (i.e. only one solution) and solutions are said to be consistent.

$\therefore a_1{b}_2-a_2{b}_1\ne 0\Rightarrow \frac{a_1}{b_2}\ne \frac{b_1}{b_2}$

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.

$\therefore a_1{b}_2-a_2{b}_1\ne 0\Rightarrow \frac{a_1}{a_2}\ne \frac{b_1}{b_2}$

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). $Px-y=2$ $6x-2y=3$

LINEAR EQUATIONS IN TWO VARIABLES I

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

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

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

$\Rightarrow \frac{P}{6}\ne \frac{+1}{+2}\Rightarrow P\ne \frac{6}{2}\Rightarrow P\ne 3\therefore P$ 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

$kx+4y=k-4$

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

LINEAR EQUATIONS IN TWO VARIABLES I

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

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

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

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

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

$\therefore k=8$ 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.

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

LINEAR EQUATIONS IN TWO VARIABLES I

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

$a_2=k^2+1,b_2=k-2\text{ and }{c}_2=-5$ For no solution, condition is $\frac{a_1}{a_2}=\frac{b_1}{b_2}\ne \frac{c_1}{c_2}$

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

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

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

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

$\Rightarrow k=-1$ Clearly, $\frac{3}{k-2}\ne \frac{2}{5}$ 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 $3x-5y+2=0$, and $6x+4=10y$ have :

(A) No solution $$

(B) A single solution

(C) Two solutions $$

(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 $3x+2y=1$ then is also satisfies :

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

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

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

(D) None of these.

LINEAR EQUATIONS IN TWO VARIABLES I

LINEAR EQUATIONS IN TWO VARIABLES I

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

(A) 1 $$

(B) 2

(C) 3 $$

(D) 4

LINEAR EQUATIONS IN TWO VARIABLES I

LINEAR EQUATIONS IN TWO VARIABLES I

4. The system of linear equation $ax+by=0,cx+dy=0$ has no solution if :

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

(B) ad - bc $<0$

(C) $ad+bc=0$ $$

(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 $kx+3y=7$ and $2x-5y=3$ has no solution is :

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

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

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

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

LINEAR EQUATIONS IN TWO VARIABLES I

LINEAR EQUATIONS IN TWO VARIABLES I

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

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

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

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

(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) $x=8,y=6$ $$

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

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

(D) None of these

LINEAR EQUATIONS IN TWO VARIABLES I

LINEAR EQUATIONS IN TWO VARIABLES I

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

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

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

(C) $k\ne 6$ $$

(D) $k=6$

LINEAR EQUATIONS IN TWO VARIABLES I

LINEAR EQUATIONS IN TWO VARIABLES I

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

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

(B) Are consistent and have infinitely many solution

(C) are inconsistent $$

(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) $y-x$ $$

(B) $x-y$

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

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

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

LINEAR EQUATIONS IN TWO VARIABLES I

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

LINEAR EQUATIONS IN TWO VARIABLES I

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

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. $\frac{x}{3}+y=1.7$ and $\frac{11}{x+\frac{y}{3}}=10\mathrm{\forall }[x+\frac{y}{3}\ne 0]$

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 $\frac{7}{x}-\frac{4}{y}=5(x\ne 0,y\ne 0)$ satisfy both the equation $2(\sqrt{3}x+4)-3(4y-5)=5$ & $7(\frac{9x}{\sqrt{3}}+8)+5(7y-25)=64$

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. $2x+3y=7$; $(a-b)x+(a+b)+b-2$

LINEAR EQUATIONS IN TWO VARIABLES I

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

LINEAR EQUATIONS IN TWO VARIABLES I

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

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: $\frac{bx}{a}-\frac{ay}{b}+a+b=0;bx-ay+2ab=0$

LINEAR EQUATIONS IN TWO VARIABLES I

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

LINEAR EQUATIONS IN TWO VARIABLES I

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

LINEAR EQUATIONS IN TWO VARIABLES I

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

LINEAR EQUATIONS IN TWO VARIABLES I

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

$x-22y-2z=5$

$2x+y-3z=1$

LINEAR EQUATIONS IN TWO VARIABLES I

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

LINEAR EQUATIONS IN TWO VARIABLES I

11. Solve, $px+qy=r$ and $qx=1+r$

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. $$ (B) becomes consistent.

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

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

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

$2kx+6y=6$ $$ $2x+ky+14=0$

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) $2xky+k+2=0$ $$ (ii) $Cx+3y=3$

$kx+8y+3k=0$ $$ $12x+Cy=6$

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 $y$ : $(a-b)x+(a+b)y=a^2-2ab-b^2$

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

[CBSE - 2008]

LINEAR EQUATIONS IN TWO VARIABLES I

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

LINEAR EQUATIONS IN TWO VARIABLES I

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

$37x+43y=123$

$43x+37y=117$ $$

[CBSE - 2008]

LINEAR EQUATIONS IN TWO VARIABLES I

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