Demensions of a matrix

this is a 1 x 3 matrix because there is 1 row and 3 columns.

this is a 3 x 3 because there are 3 rows and 3 columns.

Error Analysis

in problem 20, the line should be dotted.
in problem 21, the shading should be below the lines.

in problem 22, the line should be dotted because it is not equal to.
in problem 23, the shading should be above the line, no below it.

absolute value equations

the equation is:
y = a | x-h | + k

Isolate the absolute value
If the number on the other side of the equation negative, then it is no solution.
If the number on the other side of the equation is not negative, write two equations without absolute value bars, but have one where the number the equation equals to, be negative, and the other postive.
Then solve both equations.

|3x - 6| - 9 = -3

|3x - 6| = 6

3x - 6 = 6          3x - 6 = -6

3x = 12       3x = 0

x = 2          x = 0

Graphing absolute value equations:

the parent function for graphing absolute values is y = | x |

If the equation was y = | x + 2 | , this is how it would look:

If there was a negative outside of the absolute value sign, then it would open downwards, and look something like this:    y = - | x |

systems of linerar equations

Independent Consistant:
a equation is consistant if there is at least one solution.
a equation is independent when 2 lines cross at one point

for example:
y = x + 10
y = 2x

Dependent Consistant:
when 2 lines have the same slope and y-intercept.
one overlaps the other

for example :
y = x + 10
2y = 2x + 20



 
 
Incosistant :
this means the 2 lines do not cross at any point.
they are most likely parallel.
they do have the same slope.
 
example:
y = x
y = x + 10