Question

Find the value of a and b for which the given system of linear equation has an infinite number of solutions $$2x + 3y = 7,(a - b)x + (a + b)y = 3a + b - 2$$

Answer 1

Hint: We know the general equations i.e.
  $ {{a_1}x + b{_1}y = {c_1}}$
  $ {{a_2}x + b{_2}y = {c_2}} $
After comparing general equations with the given equation we got the value. After that we need to apply the condition of infinite solution which is $$\dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}} = \dfrac{{{c_1}}}{{{c_2}}}$$ . after this we can get the value of a and b.

Complete step-by-step answer:
Comparing $$2x + 3y = 7\;$$with $${a_1}x + {b_1}y + {c_1} = 0$$,
we get,
$${a_1} = 2,{b_1} = 3,{c_1} = 7$$
Comparing $$\left( {a - b} \right)x + \left( {a + b} \right)y = 3a + b - 2\;$$with $${a_2}x + {b_2}y + {c_2} = 0$$ ,
we get,
$${a_2} = \left( {a - b} \right),{b_2} = \left( {a + b} \right),{c_2} = 3a + b - 2$$
Given system of equation have infinite solution:
$$\eqalign{
  & \dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}} = \dfrac{{{c_1}}}{{{c_2}}} \cr
  & \Rightarrow \dfrac{2}{{a - b}} = \dfrac{3}{{a + b}} = \dfrac{7}{{3a + b - 2}} \cr
  & \cr} $$

From $$\dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}}$$​​

⇒$$\dfrac{{2}}{{a - b}} = \dfrac{{3}}{{a + b}}$$
$$\eqalign{
  & \Rightarrow 2a + 2b = 3a - 3b \cr
  & \Rightarrow a - 5b = 0.........\left( I \right) \cr} $$
Now from $$\dfrac{{{b_1}}}{{{b_2}}} = \dfrac{{{c_1}}}{{{c_2}}}$$
$$\eqalign{
  & \Rightarrow \dfrac{{3}}{{a + b}} = \dfrac{7}{{3a + b - 2}} \cr
  & \Rightarrow 9a + 3b - 6 = 7a + 7b \cr
  & \Rightarrow 9a - 7a + 3b - 7b = 6 \cr
  & \Rightarrow 2a - 4b = 6......\left( {II} \right) \cr} $$
Now, for subtracting eq I and eq II
$$\eqalign{
  & a - 5b = 0 \cr
  & 2a - 4b = 6 \cr} $$


Multiplying eq I by 2
$$\eqalign{
  & 2a - 10b = 0 \cr
  &\Rightarrow - 2a + 4b = - 6 \cr
  & \Rightarrow - 6b = - 6 \cr
  & \Rightarrow b = 1 \cr} $$
Now, substituting b value in eq I
$$\eqalign{
  & \Rightarrow 2a - 10\left( 1 \right) = 0 \cr
  &\Rightarrow 2a - 10 = 0 \cr
  &\Rightarrow 2a = 10 \cr
  & \Rightarrow a = \dfrac{{10}}{2} \cr
  & \Rightarrow a = 5 \cr
  & \Rightarrow a = 5,b = 1 \cr} $$

Note: We knew the general equations , here we compare those with equations given in problem. After that applied the condition of infinite solution which is $$\dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}} = \dfrac{{{c_1}}}{{{c_2}}}$$ . After comparing the equations we got the value of a and b.