Algebra 2 LAP 3

Inconsistent system

An inconsistent system has no solution.The slopes of the lines are the same and the y-intercepts are different. { }

To solve a system of equations:

find the solution(s) to all equations in the system, this is the set of ordered pairs that represent the intersection of the graphs of the equations.

If the line is vertical

there is no slope, or the slope is undefined./ m= (0 with a line through it)

Solving matrices

put it in standard form Ax=By=C, order matters coefficients: [A]= [A1 B1 /constants: [B]= [c1 A2 B2] c2]

If the line is horizontal

the slope is zero./ m=0

P(5, 2), m=0

horizontal line

Inverse matrix

-If the product of two square matrices, [M] and [N] is the identity matrix, then [M] and [N] are inverses of each other: [M]-1=[N] and [N]-1=[M].-Some square matrices do not have inverses. These matrices are called singular. Matrices with inverses are called invertible.

If you have a system of 2 linear equations in 2 variables, how might the graphs of those lines be oriented? (in solution sets)

-Intersect: {(x,y)} one solution -Parallel lines: { } no solutions -Coinside: {(x,y): Ax +By= C} all real numbers

To prepare for problem solving with linear systems, we need to define some terms related to air speed and speed in water:

-air speed: speed in still air (represented by "a") -boat speed: speed in still water (represented by "b")

Matrix in algebra

-an array of numbers written in rows + columnsEx: [ 5 -2 6]-3 0 -1[ -2 4 8 ]-This matrix has 3 rows and 3 columns. The order (m X n) of a matrix is the number of (m) times the number of columns (n).-The order of the matrix above is 3 X 3.

Linear inequality in 2 variables

-an open sentence in the form Ax + By > C or Ax + By < C-The graph of a linear inequality in 2 variables is a half plane (shaded region) with the graph of the associated linear equation Ax + By = C as the boundary of the half-plane.

speed of current: speed in which water is flowing (represented by "c")

-downstream: water moving in the same direction as the -boat upstream: water moving in the opposite direction as the boat

How can you tell if a relation is a function?

-from a graph use vertical line test -from a set of ordered pairs (relation) there should be no repeated x values

headwind and tailwind

-less # slow you down - wind/ against wind -go faster + wind/with wind

Lesson 26

-see if there are restrictions in a problem by setting a denominator to equal 0. -Use FOIL in some problems

wind speed: speed of the wind (represented by "w")

-tail wind: wind blowing in the direction of the aircraft -head wind: wind blowing opposite the direction of the aircraft

We identify a linear system in 2 variables by the solutions it has:

1) Consistent system= one solution. 2) Inconsistent system= no solution. 3) Dependent system= has an infinite set of solutions (the entire set of points on the line makes up the solution set.)

We will be solving a system of linear equations in 3 variables by one of two methods:

1) Create an upper triangular system and use back substitution (work up) (triangle) Ax+By+Cz=D By+Cz=D Cz=D 2) Create three matrices and use the matrix feature of the graphing calculator. (This only works if the matrix has an inverse)

Theorem: Given lines L1 and L2 with slopes m1 and m2,

1) L1 II L2 if and only if m1=m2 2) L1 (perpendicular sign upside down T) L2 if and only if m1xm2=-1

To solve a system of linear equations by using matrices: (not really important)

1) Write a square matrix [A] that contains the coefficients of the variables in the system. Enter into the calculator in the matrix menu under [A]. 2) Write a matrix [B] that contains the constants of the system. Enter into the calculatorin the matrix menu under [B]. 3) Write a matrix [X] which contains the variables of the system.So, [A] [X] = [B] 4) Solve by multiplying the inverse of [A] by [B] So, [X] = [A]-1[B]

describing a function:

1) a set of ordered pairs for which no x value is repeated Ex.: { (2, 3), (2, 5), (3, 5), (1, 2), (0, 3), (1, 4)} 2) a graph of the ordered pairs for which no two points are on a vertical line (vertical line test)

The solution to the system is represented by the intersection of the 3 planes and may be:

1) a single point (X) 2) no points (parallel //) 3) an infinite set of points (same equation)

To solve by linear combination:

1) multiply one equation by one value and the other equation by another value, so that you end up with opposites for either the "x" or "y" terms 2) add the equations together to cancel the "x" or "y" terms and solve for the other variable 3) replace that variable in one of the equations with the value you get and solve for the other variable

To graph a non vertical line on a graphing calculator,

1) put the equation into slope intercept form (y=mx+b) 2) put the equation into the calculator in the "y =" (2nd graph= T chart) menu button.

To graph a line using the slope:

1) put the equation into slope intercept form (y=mx+b), if possible, 2) plot the y-intercept, (0,b) 3) count the slope to plot a second point, 4) draw the line through the 2 points.

To graph a linear inequality

1) solve for "y" 2) graph the boundary as a dotted (open half-plane: > <) or solid (closed half-plane) line. ( _> <_) 3) shade the region above ( > or _> ) or below ( < or _< ) 4) if there is no "y" term, graph the vertical line as dotted or solid and shade to the right or left

Solving systems by substitution:

1) solve one of the equations for one of the variables 2) substitute that expression in the other equation 3) use that value to solve for the other variable

To find the slope of a line,

1) use the coordinates of two points and the formula (m= y2-y1/x2-x1) 2) use the graph of the line and count the slope as rise and run from one point to another on the line. 3) use the equation of the line: a) if the equation is in standard form Ax + By = C, then m= -A/B b) if the equation is in slope intercept form y = mx +b, then m is the slope.

To find the equation of a line:

1) use the pointslope form of the linear equation 2) substitute the slope and a point into the equation 3) convert the equation to either slopeintercept or standard form

How many graphs are needed to form a line?

2

Calculator buttons in order for matrices

2nd x^-1 1 x^-1 2nd x^-1 2 ent.

X coordinates same slope

= (0 with line through it) undefined (problem only x no slope undefined)

B in a problem represents

= (equal sign), is constant

Y coordinates same slope

= 0 horizontal

Solution to a system in 2 or 3 variables

?

boundary

?

consistent

A system that has a solution. The slopes of the lines are different. {(x,y)} Ex: y= 1/2x - 3 and y= 4/3x-4

Dependent

A system that has an infinite set of solutions (the entire set of points on the line makes up the solution set.)The slopes and y-intercepts of the lines are the same. {(x,y): Ax+By=C}

standard form

Ax + By = C

Neither (lesson #18)

Diagonal

In calculator for matrices

ERR:Singular Mat= no solution --> Inconsistent system

parallel sign

II

Open half-plane

If the linear inequality contains one of the symbols > or <, then the boundary is not included and the graph of the inequality is an open halfplane.

closed half-plane

If the linear inequality contains one of the symbols or, the boundary is included and the graph of the inequality is a closed halfplane.

Domain

One of the variables (the independent variable) in the open sentence (usually x) (x,y= solution set)

Trying to get a decimal to a fraction on a calculator (.4)

Put .4 then hit math enter enter

Coordinates

The ordered pair of numbers corresponding to a point in the coordinate plane.

Range

The other variable (the dependent variable) in the open sentence (usually y) (x,y= solution set)

origin

The point at which the 2 coordinate axes meet- its coordinates are (0,0)

The graph equation in the form Ax+By=C (A and B are not both zero) is a line.

This form of the equation is called the standard form of the linear equation

L22

When 0=3x --> { } empty set

Order of matrices ex

[0 0 0 0] 0 0 0 0 [0 0 0 0] <--- order= 3 x 4

Function

a correspondence between two sets, D (domain) and R (range) that assigns each element in D with exactly one member of R

Mapping

a mapping diagram from one set to another, like a t-chart: x values on one side the y values on the other side.

Constant

a number (on its own, don't change)

The graph of a linear equation in 3 variables is

a plane in space. (gives depth, forward and backward (intersection)

coordinate plane (Cartesian)

a rectangular coordinate systems in which there are 2 axes (one horizontal, one vertical) which divide the plane into 4 regions

System of equations or inequalities

a set of equations or inequalities in conjunction (find the intersection)

vertical line test

a visual way to determine if a curve is a graph of a function or not. No x values should repeat, if it is a function . Vertical line cannot pass vertical line test.

A linear equation in 3 variables is

an open sentence in the form Ax + By +Cz = D. D is a constant

A solution to a linear equation in 3 variables is

an ordered triple. {(x,y,z)}

Relation

any set of ordered pairs (x, y)

f(x) = b

constant function

>

dashed line, shade above

<

dashed line, shade below

x>1

dashed line, shade right

P(-2, 3); no slope

diagonal line

In some functions the

domain (usually all real #'s) is restricted because of the algebraic expression it contains. This is the implied domain of the function. Find the domains of these functions by identifying the restrictions in the denominator and/or in the radicand (anything under the radical)

Curve happens in a graph when

exponent is increased

function notation

f(x)=2x-1 f(3)=2(3)-1 f(3)=6-1 f(3)=5

T-chart

find all pairs that work together

linear function:

function whose graph is a line and that is written in the form f(x) = mx + b

mapping notation

g: x-->2x (arrow=yields) Ex: g: 3--> 2(3) g: 3--> 6

To graph a system of linear inequalities,

graph the inequalities separately and indicate the intersection of the shaded regions.

plotting a point

graphing a point in a plane by locating its coordinates in the plane

f(x) = x

identity function

Graph of solution set for an open sentence in 2 variables is

in a plane (4 quadrants, x and y axis)

Open sentences in two variables

is an equation or inequality that contains only 2 variables.

graph of an open sentence in two variables

is the set of all points in a plane whose coordinates satisfy the equation or inequality

Which ever points you choose to graph for that line

it is a good idea to write their coordinates or to put them into a T-Chart (if ask to graph he wants to see T-chart)

If the slope is a negative number

it slopes down from left to right

Long equation solve by

learning by the numbers (look at notes L22)

A system of linear equations in 2 variables can be solved by using one of the following methods:

linear combination substitution graphing Cramer's Rule matrices

opposite reciprocal

m1xm2 (m=2, reciprocal would be m= -1/2)

If you can in standard form

make A a positive number not a negative number

Standard form=

no fractions

Negative variable

no linear (lesson #18)

quadrant

one of the 4 regions in a coordinate plane

A square matrix has

order m X m (the same number of rows and columns).

g(0) = 5, an increase of 4 units in x causes a decrease of 12 units for g(x)

rate of change m=y/x, m=-12/4, m=-3

slope is interpreted as the

rate of change: m = change in f(x)/ change in x (m= triangle shape times y over triangle shape times x.) triangle= delta, change

A system of linear equations in 3 variables is

represented by three planes

(5, -3) and (5, 4)

same x coordinate, vertical line, x=5

(-2, 5) and (3,5)

same y coordinate, horizontal line, y=5

_<

solid line, shade above

_>

solid line, shade above

y_>-2

solid line, shade above

ordered pair

solution to open sentence in 2 variables

If C is zero

solve the equation for "y" and pick any two values for x then find y. Use a third point as a "check".

identity matrix:

square matrix in which all entries are "0" except the main diagonal (diagonal going from the upper left corner to the lower right corner) for which the entries are all "1".

Coeffiicient

the constant (or numerical) factor in a variable

Abscissa

the first coordinate in an ordered pair (x in (x,y)) (x-coordinate: the first number in an ordered pair)

If the equation is in the form y=b

the graph of the equation is a horizontal line

if the equation is in the form x=a

the graph of the equation is a vertical line

axes

the horizontal and vertical coordinate lines which divide the coordinate plane

x-axis

the horizontal number line in a coordinate plane

If the slope is a positive number

the line slopes up from left to right

Ordinate

the second coordinate in an ordered pair (y in (x,y)) (y-coordinate: the second number in an ordered pair)

Relation

the solution set for an open sentence in 2 variables is a set of ordered pairs (x,y)

If C is not zero

the two points you graph can be the x-intercept (the point where the graph crosses the x-axis) whose coordinates are (x, 0) and the y-intercept (the point where the graph crosses the y-axis ) whose coordinates are (0, y).

The slope (m) of a line in a plane is the ratio of

the vertical change (rise) to the horizontal change (run) as you move from one point to another on the line. (rise/run)

y-axis

the vertical number line in a coordinate plane

x-intercept

the x-coordinate of a point where a graph crosses the x-axis

y-intercept

the y-coordinate of a point where a graph crosses the y-axis

Linear system

two or more linear equations that have the same variables (linear equations: straight line)

Perpendicular sign

upside down T

Perpendicular problems

use opposite reciprocal m=B/A

Parallel problems

use same slope m=-A/B

If a system is dependent (infinitely many solutions),

we write the general solutions in terms of z.

You can find the range of a function (y-values)

when given the values from the domain (x-values) and a function rule in the form of a mapping or function notation (no solution sets)

ground speed: speed relative to the ground

with a tail wind: a + w with a head wind: a - w downstream: b + c upstream: b - c

point-slope form

y - y1 = m(x - x1)/ slope=m and point: (x1,y1)

slope intercept form

y = mx + b/ slope= m and y-intercept: (0, b)

Putting things in solution sets is always alphabetically:

{(x,y)} --> {(-2/3,0)}

r(x)= x^2 + 5

{x: all real numbers}