Question

Find the value of ‘p’ and ‘q’ for which the following pair of linear equations has infinite number of solutions. 2x + 3y = 7 , 2px +(p + q)y = 28

Answer 1

Answer:

For infinite many solution,

$\frac{a1}{a2} = \frac{b1}{b2} = \frac{c1}{c2}$

now,

$\frac{2p}{2} = \frac{p + q}{3} = \frac{28}{7 } = 4$

$2p = 8 \: \: i.e \: p = 4$

$p + q = 12 \: \: so \: 4 + q = 12 \: \: i.e \: \: q = 8$

Hence P = 4 & Q = 8. Ans.

Answer 2

$\huge\bf{Answer:-}$

Given:-

The equations are -

• 2x + 3y = 7
• 2px + ( p + q )y = 28

To Find :-

The value of p and q

Solution:-

2x + 3y = 7

2px + ( p + q )y = 28

Now, Comparing both the equations to the standard form ax + by + c = 0 we get,

2x + 3y -7 = 0 ......(i)

2px + ( p + q )y - 28 = 0 .....(ii)

Here , in eq (i)

a1 = 2 , b 1= 3 , c1 = -7

In eq(ii)

a 2= 2p , b 2= (p + q) , C2 = - 28

For infinitely many solution ,

⇒$\frac{</strong><strong>a</strong><strong>1</strong><strong>}{</strong><strong>a</strong><strong>2</strong><strong>} </strong><strong>=</strong><strong> </strong><strong>\frac{</strong><strong>b</strong><strong>1</strong><strong>}{</strong><strong>b</strong><strong>2</strong><strong>}</strong><strong> </strong><strong>=</strong><strong> </strong><strong>\</strong><strong>f</strong><strong>r</strong><strong>a</strong><strong>c</strong><strong>{</strong><strong>c</strong><strong>1</strong><strong>}</strong><strong>{</strong><strong>c</strong><strong>2</strong><strong>}</strong><strong>$

⇒$\frac{2}{2p} = \frac{3}{p+q} =\frac{-7}{-28}$

so,

⇒$\frac{2}{2p} = \frac{3}{p+q}$

Now , by cross multiplication ,

⇒2p + 2q = 6p

⇒2p - 6p = -2q

⇒-4p = -2q

⇒2p = q ..... ( iii)

Now,

$\frac{3}{p+q} = \frac{-7}{-28}$

Again, By cross multiplication

7p + 7q = 84

p+ q = 84 /7

p + q = 12 .....( iv)

Now putting the value of q that we find out in eq (iii) in eq( iv) , we get

p+ q = 12

p + 2p = 12

3p = 12

p = 12 / 3

p = 4

Therefore, the value of p is 4

Now , we get 2p = q in eq (iii)

so , putting the value of p in eq (iii)

2 × 4 = q

q = 8

Hence the value of p is 4 and q is 8 .