Download Creating an Inconsistent system of linear equations

ALEKS PURPLE Creating an Inconsistent System of Linear
Equations
For these problems, you will be given a system of linear equations (both in
standard form, fortunately). The coefficient of one of the variables in one of the
equations will be written as the letter-b (or some other letter of the alphabet). You
are asked to find out what value of b would make this system of linear equations
inconsistent.
A system of linear equations is inconsistent when there is no solution. There
is no solution when you graph the lines and the lines are parallel. Thus, you can
find the value of the letter-b if you can make the slope of the two equations
equal (remember, parallel lines have the same slope).
Remember how to find the slope? When an equation is written in slope-intercept
form or in point-slope form, the slope is the coefficient of the x variable. However,
when the equation is written in standard form, the slope is the coefficient of the x
variable divided by the opposite (negative) of the coefficient of the y variable.
Thus, if the equation is 10x - 15y = 7, the slope is 10 divided by 15 (the coefficient
of the y-variable is -15 so we divide by positive 15 to find the slope). 10/15 = 2/3
If the equation is 12x + 24y = 6, the slope is 12 divided by -24 = 12/-24 = -1/2
So let's look at a sample problem:
Find the value of b such that the system of equations
-16x - 20y = -7
12x + by = 16
does not have a solution.
See if you can solve this problem on your own before looking at the next page.
The slope of the first line is -16/20 or -4/5
The slope of the second line is 12/-b
If this system is inconsistent, the two slopes must be the same:
-4/5 must be the same as 12/-b, which is the same as -4/5 = -12/b