Question

Show that the equations `9x - 10y = 21, (3x)/2 - (5y)/3 = 7/2` have infinitely many solutions.

Answer 1

1. Write equations in standard form

Express both linear equations in the standard form ax + by + c = 0:

First Equation (E₁):

9x – 10y – 21 = 0

Here, a₁ = 9, b₁ = –10 and c₁ = –21.

Second Equation (E₂):

`(3x)/2 - (5y)/3 - 7/2 = 0`

Here, `a_2 = 3/2, b_2 = -5/3` and `c_2 = -7/2`.

2. Calculate the coefficient ratios

Find the individual ratios for the coefficients of x, y and the constant terms:

Ratio of x-coefficients `(a_1/a_2)`:

`(a_1)/(a_2) = 9/(3/2)`

= `9 xx 2/3`

= 6

Ratio of y-coefficients `(b_1/b_2)`:

`(b_1)/(b_2) = (-10)/(-5/3)`

= `-10 xx (-3/5)`

= 6

Ratio of constants terms `(c_1/c_2)`:

`(c_1)/(c_2) = (-21)/(-7/2)`

= `-21 xx (-2/7)`

= 6

3. Compare the ratios

Evaluate whether the calculated ratios satisfy the condition for infinite solutions:

`(a_1)/(a_2) = (b_1)/(b_2) = (c_1)/(c_2) = 6`