Equations of the Forms - Linear Equations in Two Variables, Class 10 Mathematics PDF Download

• EQUATIONS OF THE FORM ax + by = c AND bx + ay = d, WHERE a ¹ b.

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To solve the equations of the form :
ax + by = c                                                                     ...(i)
and bx + ay = d                                                             ...(ii)
where a ¹ b, we follow the following steps :
Step-I : Add (i) and (ii) and obtain (a + b)x + (b + a) y = c + d, i.e., x + y = c d a b +
+ ...(iii)
Step-II : Subtract (ii) from (i) and obtain (a – b)x – (a – b) y = c – d, i.e., x – y = c – d a – b ...(iv)
Step-III : Solve (iii) and (iv) to get x and y.

Ex.12 Solve for x and y : 47x + 31y = 63,  31x + 47y = 15.
Sol. We have, 47x + 31y = 63  ...(i)  and 31x + 47y = 15 ...(ii) Adding (i) and (ii), we get : 78x + 78y = 78 Þ x + y = 1 ...(iii) Subtracting (ii) from (i), we get : 16x – 16y = 48 Þ x – y = 3 ...(iv) Now, adding (iii) and (iv), we get : 2x = 4 Þ x = 2 Putting x = 2 in (iii), we get : 2 + y = 1 Þ y = – 1 Hence, the solution is x = 2 and y = –1

• EQUATIONS REDUCIBLE TO LINEAR EQUATIONS IN TWO VARIABLES

Equations which contain the variables, only in the denominators, are called reciprocal equations. These equations can be of the following types and can be solved by the under mentioned method :

Type-I : c' ∈ a,b,c,a',b',c'∈ R

Put 1
x
u = and 1
v
y = and find the value of x and y by any method described earlier.
Then u = 1
x  and v = 1
y Type-II : au + bv = cuv and a'u + b'v = c'uv " a,b,c,a',b',c'ÎR Divide both equations by uv and equations can be converted in the form explained in (I).

Type-III : a b k lx m y cx dy + =
+ + , 
a ' b ' k ' lx m y cx dy + =
+ +
" a,b,k,a',b',k'ÎR
Put 1
u lx m y =
+
 and 
1
v cx dy =
+
Then equations are au + bv = k and a'u + b'v = k'
Find the values of u and v and put in Ix + my = 1
u  and cx + dy = 1
v Again solve for x and y, by any method explained earlier.