Question

Muriel sees she has written a system of two linear equations that has an infinite number of solutions one of the equations of the system is 3Y = 2X -9 what could be the other equation?

Answer 1

Eq2 : 6Y = 4X - 18

Explanation:

- For any system of equation to have "infinitely" many solutions then at-least 2 equations must be dependent equations.

- The dependent equations have many solutions as the existence of one makes the other equation redundant,

- In other words, the basic example would be a scalar multiple of original equation.

aY = bX + C ..... Eq 1

- The scalar multiple "k", can be any non-zero real value:

kaY = kbX + Ck .... Eq 2

- We divide the two equations:

Eq2 / Eq1 = k ... constant (non-zero)

- Linear independence check invalidates as their exist no non-zero scalar multiple (a,b) such that:

Eq1*a + Eq1*b= 0. ..... Hence, equations are dependent

- So choose any value of k = 2:

2*Eq1 : Eq2

Eq2 : 2*(3Y = 2X -9)

Eq2 : 6Y = 4X - 18

Answer 2

Then the other equation can be 6Y = 4X - 18

Explanation:

Here, where there are two equations representing the same line, the number of solutions is infinite.

That is the two lines meet at all points providing an infinite number of solutions. The two lines can be said to be equivalent, such that every point on each of the two lines is a solution.

Since the equation of one member of the system is 3Y = 2X - 9

Then the other equation can be

2 × (3Y = 2X - 9) → 6Y = 4X - 18.