10th Class Mathematics Pair of Linear Equations in Two Variables Elimination Method of Solving

       Algebraic Method of Solving the System of Equation

There are different methods of solving the system of linear equations. The three different methods are:

(a) Elimination Method

(b) Substitution Method

(c) Cross Multiplication Method

       Elimination Method

In this method first we eliminate one of the variables by equating the coefficient of the one of the variable and finding the other variable. Then again re-substituting the value and getting the value of other variable.    

Solve the system of the equation given by \[\frac{4}{16x+24z}+\frac{12}{21x-14z}=\frac{1}{2}\] and \[\frac{14}{4x+6z}+\frac{4}{(3x-2z)}=2\]

(a) \[(x=2,z=1)\]                                 

(b) \[(x=3,z=5)\]

(c) \[(x=-4,z=1)\]                                

(d) \[(x=1,z=-1)\]                            

(e) None of these

Answer: (a)

The value of x and y which satisfies the system of equation \[ax+ry=p+q\] and \[\left( \frac{p}{p-q}-\frac{p}{p+q} \right)x+\left( \frac{r}{q-p}-\frac{r}{q+p} \right)=\frac{2p}{p+q}\]

(a) \[\left( x-\frac{q}{r},y=\frac{p}{q} \right)\]                      

(b) \[\left( x-\frac{r}{p},y=\frac{p}{q} \right)\]

(c) \[\left( x=\frac{p}{q},y=\frac{q}{r} \right)\]                     

(d) \[\left( x=-\frac{p}{q},y=\frac{r}{p} \right)\]

(e) None of these

Answer: (c)

Robert and Smith have certain number of cakes. Robert says to Smith/if you give me 10 of your cakes, I will have twice the number of cakes left with you. Smith replies that if you give me 10 of yours cakes then I will have the same number of cakes as left with you. Find the number of cake with Robert and Smith respectively.

(a) (45, 65)                                             

(b) (70, 50)

(c) (80, 60)                                             

(d) (60, 40)

(e) None of these

Answer: (b)

Explanation

Let the number of cake with Robert be x and that with Smith be y.

Then by first condition,

\[x-2y=-30-----(1)\]

By second condition,

\[x-y=20-----(2)\]

On solving the above equation we get,

\[x=70\,and\,y=50\]

Mary sells a washing machine at the gain of 10% and trolley at the gain of 5% and earns a profit of Rs.3000. But if she sells the washing machine at the gain of 5% and trolley at the loss of 10%, she gains 1000. Find the cost price of washing machine and trolley respectively.

(a) (Rs. 14000, Rs. 2000)                   

(b) (Rs. 1400, Rs. 2000)

(c) (Rs. 10000, Rs. 6000)                   

(d) (Rs. 3000, Rs. 8000)

(e) None of these

Answer (a)

Thomas thinks of a two number such that if the larger of the two numbers is divided by the smaller one, he gets remainder as 6 and quotient as 1. At the same time if three times of smaller number is divided by the larger one gets quotient as 2 and remainder as 12. The number he thought of is.

(a) (12, 20)                                             

(b) (30, 24)

(c) (30, 16)                                             

(d) (18, 24)                         

(e) None of these

Answer: (b)

         Substitution Method

In this method we find the value of one variable in terms of other and then substitute the value of the variable in the second equation and solve and get the value of the first variable. Again we substitute the value of the variable obtained in first step and in the previous equation and get the value of the other variable.

        Cross Multiplication Method

In this method we translocate the coefficient of both the equation by cross multiplication with the coefficient of other equation and get the value of the variable.

For the system of linear equation,

             \[{{a}_{1}}x+{{b}_{1}}y+{{c}_{1}}=0.......(i)\]

             \[{{a}_{2}}x+{{b}_{2}}y+{{c}_{2}}=0.......(ii)\]

             \[x\left( \frac{{{b}_{1}}{{c}_{2}}-{{b}_{2}}{{c}_{1}}}{{{a}_{1}}{{b}_{2}}-{{a}_{2}}{{b}_{1}}} \right),\,and\,y=\frac{\left( {{c}_{1}}{{a}_{2}}-{{c}_{2}}{{a}_{1}} \right)}{({{a}_{1}}{{b}_{2}}-{{a}_{2}}{{b}_{1}})}\]