# How do you solve #-4x – 6y = -28# and #2x + 3y = 14 #?

##### 1 Answer

Here's what I got.

#### Explanation:

Your starting system of two equations with two unknowns looks like this

#{(-4x - 6y = -28), (color(white)(..)2x + 3y = color(white)(-)14) :}#

Notice that if you multiply the second equation by

#2x + 3y = 14 " "| xx color(blue)((-2))#

#color(blue)((-2)) * 2x + color(blue)((-2)) * 3y = color(blue)((-2)) * 14#

#-4x -6y = -28#

This means that your system of equations has an **infinite number of solutions**. This is the case because if you were to subtract this new form of the second equation from the first equation, you'd get

#{(-4x - 6y = -28" "| -), (-4x -6y = -28) :}#

#color(white)(aaaaaaa)/color(white)(aaaaaaaaaaaaaaaaa)#

#-4x - (-4x) - 6y - (-6y) = -28 - (-28)#

#-color(red)(cancel(color(black)(4x))) + color(red)(cancel(color(black)(4x))) - color(red)(cancel(color(black)(6y))) + color(red)(cancel(color(black)(6y))) = -color(red)(cancel(color(black)(28))) + color(red)(cancel(color(black)(28)))#

which of course gives

#0 = 0#

So, when does *always*, which is why your system of equations is said to have and **infinite** number of solutions.

In other words, you can plug in any value you want for **always** be true.

Alternatively, you can think about the two equations given to you as *describing the same line*. To check that this is the case, rearrange both equations in *slope-intercept form*

#-4x - 6y = -28#

#-6y = 4x -28 implies y = -2/3x + 14/3#

Similarly, you have

#2x + 3y = 14#

#3y = - 2x + 14 implies y= -2/3x + 14/3#

This once again leads to the conclusion that the system has an **infinite** number of solutions.