# 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?

Oct 21, 2017

$b$ would have to be $- 8$ in order for there to be infinitely many solutions to the system $3 y + 12 = 6 x$ and $2 y = 4 x + b$.

#### Explanation:

Let's review first. When does a system of equations haveinfinitely many solutions? Easy! When they're the same equation. Because when they're the same equation, they will both have identical solutions.

So, we trying to find the value for $b$ such that when plugged into the equation $2 y = 4 x + b$, it will have infinitely many solutions with the other equation $3 y + 12 = 6 x$. According to our paragraph above,

$3 y + 12 = 6 x$

and

$2 y = 4 x + b$

are the same equation.

Before we get started, let's organize the equations a little bit so that they're both in $y = m x + b$ form:

$y = 2 x - 4$

and

$y = 2 x + \frac{1}{2} b$

The $y$'s are the same, so

$2 x - 4 = 2 x + \frac{1}{2} b$

We can see that the $2 x$'s are also the same, so that leaves

$- 4 = \frac{1}{2} b$

Using simple algebra, we get

$b = - 8$