For which values of A and B will the following pair of linear equations has infinitely many solutions x 2y 1 and a B x a by a/b 2?

The given pair of linear equations are:

x + 2y = 1 …(i)

(a-b)x + (a + b)y = a + b – 2 …(ii)

On comparing with ax + by = c = 0 we get

a1 = 1, b1 = 2, c1 = – 1

a2 = (a – b), b2 = (a + b), c2 = – (a + b – 2)

a1 /a2 = 1/(a-b)

b1 /b2 = 2/(a+b)

c1 /c2 = 1/(a+b-2)

For infinitely many solutions of the, pair of linear equations,

a1/a2 = b1/b2=c1/c2(coincident lines)

so, 1/(a-b) = 2/ (a+b) = 1/(a+b-2)

Taking first two parts,

1/(a-b) = 2/ (a+b)

a + b = 2(a – b)

a = 3b …(iii)

Taking last two parts,

2/ (a+b) = 1/(a+b-2)

2(a + b – 2) = (a + b)

a + b = 4 …(iv)

Now, put the value of a from Eq. (iii) in Eq. (iv), we get

3b + b = 4

4b = 4

b = 1

Put the value of b in Eq. (iii), we get

a = 3

So, the values (a,b) = (3,1) satisfies all the parts. Hence, required values of a and b are 3 and 1 respectively for which the given pair of linear equations has infinitely many solutions.

Solution

Step 1: Convert the equations to standard formGiven two lines are,x+2y=1 and (a-b)x+(a+b)y=a+b-2 The standard linear equation is given as,ax+by+c=0Then, the two equations in standard form are given as,x+2y=1⇒x+2y- 1=0(a-b)x+(a+b)y=a+b-2⇒(a-b)x+(a+b)y-a -b+2=0Step 2: Compare the equations with the standard equationComparing the two equations with the standard equation we define,a1=1, b1=2 and c1=-1a2=a-b, b2=a+b and c2=-(a+b-2 )Step 3: Defining the condition for infinite solutionsFor infinitely many solutions of the pair of linear equations condition is satisfied by,a1a2=b1b2=c1c2 Thus,1a-b=2a+b=1a+b-2We have,1a-b=2a+b⇒2(a-b)=a+b⇒ a=2bWe also have,2a+b=1a+b-2 ⇒a+b=2(a+b-2)⇒a+b=4Substituting the above equation in the previous equation,2b+b=4 ⇒b=1Thus, a=2×1=2Therefore, when a=2 and b=1, the given set of linear equations has infinitely many solutions.

The given pair of linear equations are:

x + 2y = 1   ......(i)

(a – b)x + (a + b)y = a + b – 2   ......(ii)

On comparing with ax + by = c = 0 we get

a1 = 1, b1 = 2, c1 = – 1

a2 = (a – b), b2 = (a + b), c2 = – (a + b – 2)

`a_1/a_2 = 1/(a - b)`

`b_1/b_2 = 2/(a + b)`

`c_1/c_2 = 1/(a + b - 2)`

For infinitely many solutions of the, pair of linear equations,

`a_1/a_2 = b_1/b_2 = c_1/c_2`   .....(Coincident lines)

So, `1/(a - b) = 2/(a + b) = 1/(a + b - 2)`

Taking first two parts,

`1/(a - b) = 2/(a + b)`

a + b = 2(a – b)

a = 3b   .......(iii)

Taking last two parts,

`2/(a + b) = 1/(a + b - 2)`

2(a + b – 2) = (a + b)

a + b = 4   .......(iv)

Now, put the value of a from equation (iii) in equation (iv), we get

3b + b = 4

4b = 4

b = 1

Put the value of b in equation (iii), we get

a = 3

So, the values (a,b) = (3,1) satisfies all the parts.

Hence, required values of a and b are 3 and 1 respectively for which the given pair of linear equations has infinitely many solutions.

The given pair of linear equations are:

x + 2y = 1 …(i)

(a-b)x + (a + b)y = a + b – 2 …(ii)

On comparing with ax + by = c = 0 we get

a1 = 1, b1 = 2, c1 = – 1

a2 = (a – b), b2 = (a + b), c2 = – (a + b – 2)

a1 /a2 = 1/(a-b)

b1 /b2 = 2/(a+b)

c1 /c2 = 1/(a+b-2)

For infinitely many solutions of the, pair of linear equations,

a1/a2 = b1/b2=c1/c2(coincident lines)

so, 1/(a-b) = 2/ (a+b) = 1/(a+b-2)

Taking first two parts,

1/(a-b) = 2/ (a+b)

a + b = 2(a – b)

a = 3b …(iii)

Taking last two parts,

2/ (a+b) = 1/(a+b-2)

2(a + b – 2) = (a + b)

a + b = 4 …(iv)

Now, put the value of a from Eq. (iii) in Eq. (iv), we get

3b + b = 4

4b = 4

b = 1

Put the value of b in Eq. (iii), we get

a = 3

So, the values (a,b) = (3,1) satisfies all the parts. Hence, required values of a and b are 3 and 1 respectively for which the given pair of linear equations has infinitely many solutions.

For which values of A and B will the following pair of linear equation has infinitely many solutions?

For which values of A and B will the following pair of linear equations has infinitely many solutions x 2y 1 and a B x a b'y a/b 2?

For which values of A and B will the following pair of linear equations have infinitely many 2x 3y 7?

How do you know if a linear equation has infinite solutions?