Question

A teacher wrote the equation 3y 12 = 6x on the board. For what value of b would the additional equation 2y = 4x b form a system of linear equations with infinitely many solutions?.

Answer 1

When the given equations represent the same line, the number of points of intersection are infinite, such that the number of solutions are also infinite.

• The value of b that forms a system with infinite number of solutions is; b = -8

Reasons:

The equation the teacher wrote on the board is; 3·y + 12 = 6·x

The additional equation is; 2·y = 4·x + b

Required:

The value of b, such that the two equation form a system with infinitely many solutions.

Solution:

Two equations will have infinite number of solutions when they are the same equation, therefore, we have;

For the equation the teacher wrote;

3·y + 12 = 6·x

3·y  = 6·x - 12

y = (6·x - 12) ÷ 3 = 2·x - 4

y = 2·x - 4

For the additional equation, we have;

2·y = 4·x + b

y = (4·x + b) ÷ 2 = 2·x + b÷2

Which gives;

$\displaystyle y = \mathbf{2 \cdot x + \frac{b}{2}}$

When the two equations have infinitely many solutions, they will be equal, which gives;

$\displaystyle y = 2 \cdot x + \frac{b}{2} = 2 \cdot x - 4$

$\displaystyle 2 \cdot x + \frac{b}{2} = \mathbf{ 2 \cdot x - 4}$

$\displaystyle 2 \cdot x + \frac{b}{2} - 2 \cdot x = 2 \cdot x - 4 - 2 \cdot x; \ by \ \mathbf{subtraction \ property \ of \ equality}$

$\displaystyle \frac{b}{2} = - 4$

b = -4 × 2 = -8

b = -8

Which gives;

2·y = 4·x - 8

• The value of b for which the additional equation 2·y = 4·x + b form a system of linear equation with infinitely many solutions is b = -8.

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