Question

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For what values of ‘a’ and ‘b’ the following pairs of equation will have infinitely many solutions?
2x + 3y = 7 ;
2ax + by = 28 – by

Answer 1

Given:Two equations, 2x + 3y = 7 , 2ax + ay = 28 - by

To find: Value of a and b for which the following pair of linear equations has infinite number of solutions.

Solution:

As we have given the two equations, we first need to write the coefficients in an order.

So the coefficients should be written as if equation is Ax + By = C,

• So, in first equation:

2x + 3y = 7

A1 = 2, B1 = 3, C1 = 7

• in second equation:

2ax + ay + by = 28

A2 = 2a, B2 = a+b, C2 = 28

• Now for infinitely many solutions, we have the condition that:

A1/A2 = B1/B2 = C1/C2

• So applying this in the above equations, we get:

2 / 2a = 3/a+b = 7/28

• Now solving any two of the above, we get:

2/2a = 7/28

1/a = 1/4

a = 4

• Now solving next two from above, we get:

2/2a = 3/a+b

1/4 = 3/4+b

4+b = 12

b = 8

Answer:

So, for a = 4 and b = 8 , the following pair of linear equations has infinite number of solutions.

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Answer 2

Answer:

Given: Two equations, 2x + 3y = 7 ,   2ax + ay = 28 - by

To find:  Value of a and b for which the following pair of linear equations has infinite number of solutions.

Solution:

As we have given the two equations, we first need to write the coefficients in an order.

So the coefficients should be written as if equation is Ax + By = C,

So, in first equation:

               2x + 3y = 7

               A1 = 2,   B1 = 3,   C1 = 7

in second equation:

               2ax + ay + by = 28

              A2 = 2a,   B2 = a+b,   C2 = 28

Now for infinitely many solutions, we have the condition that:

               A1/A2 = B1/B2 = C1/C2

So applying this in the above equations, we get:

               2 / 2a = 3/a+b = 7/28

Now solving any two of the above, we get:

               2/2a = 7/28

               1/a = 1/4

               a = 4

Now solving next two from above, we get:

              2/2a = 3/a+b

              1/4 = 3/4+b

              4+b = 12

              b = 8

Answer:

                So, for a = 4 and b = 8 , the following pair of linear equations has infinite number of solutions.