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System of linear equations

Solution of a system of linear equations

Consistent system

Independent system

Two or more linear equations using the same variables.

Any ordered pair that makes all of the inequalities in the sy…

A system of equations that has at least one solution is consi…

A system of linear equations that has a unique solution.

System of linear equations

Two or more linear equations using the same variables.

Solution of a system of linear equations

Any ordered pair that makes all of the inequalities in the sy…

systems of equations

consistent system of equations

Independent system of equations

dependent system of equations

Set of equations with the same variable

A system of equations that is said to have exactly one soluti…

A consistent system of equations with exactly one solution

A system of equations that has an infinite number of solutions.

systems of equations

Set of equations with the same variable

consistent system of equations

A system of equations that is said to have exactly one soluti…

system of linear equations

consistent system

inconsistent system

dependent system

consists of two or more linear equations in the same variables

solution exists (one or more solutions)

no solution (parallel)

same graph

system of linear equations

consists of two or more linear equations in the same variables

consistent system

solution exists (one or more solutions)

Solution of an inequality

Equivalent inequalities

Roster form

Set-builder notation

Any value or values of a variable in the inequality that make…

Inequalities that have the same set of solutions.

A notation for listing all of the elements in a set using set…

A notation used to describe the elements of a set.

Solution of an inequality

Any value or values of a variable in the inequality that make…

Equivalent inequalities

Inequalities that have the same set of solutions.

Literal Equations

Complex Numbers

The Complex Number √-1

The Complex Number i^2

equations that are all or mostly all variables

a number written in the form (a + bi) where a and b are Real…

= i

= -1

Literal Equations

equations that are all or mostly all variables

Complex Numbers

a number written in the form (a + bi) where a and b are Real…

Boundary

Compound Inequality

Inequality

Intersection

A line or curve that separates the coordinate plane into regi…

Two or more inequalities that are connected by the words "and…

An open sentence that contains the symbol ˂, >, ≤, or ≥

The graph of a compound inequality containing and; the soluti…

Boundary

A line or curve that separates the coordinate plane into regi…

Compound Inequality

Two or more inequalities that are connected by the words "and…

arithmetic sequence

dependent variable

function

function rule

add the same number to the previous term (pattern)

depends on the input value (output)

a relation that assigns exactly one output value to one input…

describes the relationship between each input and output.

arithmetic sequence

add the same number to the previous term (pattern)

dependent variable

depends on the input value (output)

System of Linear Equations

Solution of a system of linear equations

Substitution Method

Elimination Method

two or more linear equations

The ordered pair that makes all the equations in a system true

When you solve one of the equations in the system for x or y,…

When you add or subtract the linear equations so that one var…

System of Linear Equations

two or more linear equations

Solution of a system of linear equations

The ordered pair that makes all the equations in a system true

first differences

solution

intersection point

compound inequality

determined by calculating the difference between successive p…

any value that makes the open sentence true

the ordered pair that satisfies both functions at the same time

an inequality that is formed by the union, "or," or the inter…

first differences

determined by calculating the difference between successive p…

solution

any value that makes the open sentence true

consistent

dependent

elimination method

inconsistent

A system of equations that has at least one solution

A consistent system that has infinitely many solutions.

If a=b and c=d, then a+c=b+d and a-c=b-d. In this method, you…

A system of equations that has no solution.

consistent

A system of equations that has at least one solution

dependent

A consistent system that has infinitely many solutions.

Addition Property of Equality

Addition Property of Inequality

coefficient

Division Property of Equality

If you add the same number to each side of an equation, the t…

If you add the same number to each side of an inequality, the…

The numerical factor of a term that contains a variable.

If you divide each side of an equation by the same nonzero nu…

Addition Property of Equality

If you add the same number to each side of an equation, the t…

Addition Property of Inequality

If you add the same number to each side of an inequality, the…

axis of symmetry

parabola

square root property

completing the square

a line about which a figure is symmetric

the set of all points in plane that are the same distance fro…

for any real number n, if x squared = n, the x = plus or minu…

a process used to make a quadratic expression into a perfect…

axis of symmetry

a line about which a figure is symmetric

parabola

the set of all points in plane that are the same distance fro…

*KeyConcepts*

Systems of Equations

Systems of Inequalities

*KeyVocabulary*

...

-a system with a graph of two intersecting lines has one solu…

-a system of inequalities is a set of two or more inequalitie…

...

*KeyConcepts*

...

Systems of Equations

-a system with a graph of two intersecting lines has one solu…

5 Properties of Inequality

The Exterior Angle Inequality Thm

Conditional

Contrapositive

1. if a > b and c>= d then a+c > b+d... 2. if a > b and c > 0 th…

the measure of an exterior angle of a triangle is greater tha…

if p then q

if not q then not p

5 Properties of Inequality

1. if a > b and c>= d then a+c > b+d... 2. if a > b and c > 0 th…

The Exterior Angle Inequality Thm

the measure of an exterior angle of a triangle is greater tha…

Inequalities

Signs of Inequalities

Types of Inequalities

Linear Inequality

The relation between two expressions that are not equal, empl…

< and > are used open circle.... ≤ and ≥ are used closed circle.

There are two types of Inequalities - Linear Inequalities and…

A Linear inequality is an inequality which involves a linear…

Inequalities

The relation between two expressions that are not equal, empl…

Signs of Inequalities

< and > are used open circle.... ≤ and ≥ are used closed circle.

Equation

Solution

Equivalent equation

Subtraction Property of Equality

A mathematical sentence that contains an equal sign, = , stat…

A replacement value for the variable in an open sentence. A v…

Two or more equations with the same solution.

If you subtract the same number from each side of an equation…

Equation

A mathematical sentence that contains an equal sign, = , stat…

Solution

A replacement value for the variable in an open sentence. A v…

5 Properties of Inequality

The Exterior Angle Inequality Thm

Conditional

Contrapositive

1. if a > b and c>= d then a+c > b+d... 2. if a > b and c > 0 th…

the measure of an exterior angle of a triangle is greater tha…

if p then q

if not q then not p

5 Properties of Inequality

1. if a > b and c>= d then a+c > b+d... 2. if a > b and c > 0 th…

The Exterior Angle Inequality Thm

the measure of an exterior angle of a triangle is greater tha…

Consistent

Dependent

Elimination Method

Inconsistent

a system of equations that has at least one ordered pair that…

a system of equations that has an infinite number of solutions

solving systems by adding or subtracting equations to elimina…

A system of equations that has no solutions

Consistent

a system of equations that has at least one ordered pair that…

Dependent

a system of equations that has an infinite number of solutions

composite function

index

inverse function

inverse relation

a function formed by evaluating one function with another fun…

the part of a radical that gives the degree of the root.

a function that "undoes" what the original function does. Swi…

the set or ordered pairs obtained by switching the x and y co…

composite function

a function formed by evaluating one function with another fun…

index

the part of a radical that gives the degree of the root.

system of linear equations

solution of the system of linear equat…

no solution

infinitely many solutions

A set of two or more linear equations containing two or more…

The intersection point of the two lines

a system of equations in which there are no solutions (the li…

The number of solutions of a system of equations in which the…

system of linear equations

A set of two or more linear equations containing two or more…

solution of the system of linear equat…

The intersection point of the two lines

consistent

dependent

elimination

inconsistent

When the graphs of a system of equations intersect or coincide.

A system of equations with an infinite number of solutions.

Solving systems of equations by adding the two equations toge…

A system of equations with no ordered pairs that satisfy the…

consistent

When the graphs of a system of equations intersect or coincide.

dependent

A system of equations with an infinite number of solutions.

Consistent

Dependent

Elimination

Inconsistent

a system of equations that has at least one ordered pair that…

a system of equations that has an infinite number of solutions.

the use of addition or subtraction to eliminate one variable…

a system of equations with no ordered pairs that satisfy the…

Consistent

a system of equations that has at least one ordered pair that…

Dependent

a system of equations that has an infinite number of solutions.

consistent

dependent

elimination

inconsistent

a system of linear equations that has at least one ordered pa…

a system of linear equations that has an infinite number of s…

the use of addition or subtraction to eliminate one variable…

a system of equations with no ordered pair that satisfies bot…

consistent

a system of linear equations that has at least one ordered pa…

dependent

a system of linear equations that has an infinite number of s…

axis of symmetry

parabola

square root property

completing the square

a line about which a figure is symmetric

the set of all points in plane that are the same distance fro…

for any real number n, if x squared = n, the x = plus or minu…

a process used to make a quadratic expression into a perfect…

axis of symmetry

a line about which a figure is symmetric

parabola

the set of all points in plane that are the same distance fro…

composite function

index of a radical

inverse function

inverse relation

a function formed by evaluating one function with another fun…

the part of a radical that gives the degree of the root.

a function that "undoes" what the original function does.

a relationship in which one variable decreases when another v…

composite function

a function formed by evaluating one function with another fun…

index of a radical

the part of a radical that gives the degree of the root.

Solve: 2n+6<4

Solve: 5n-4>11

Solve: 2x+5<6

If the inequality is 2<x, which way do…

n<-1 ... Graph: open circle at -1, shade to the left

n>3... Graph: open circle at 3, shade to the right

x<1/2 or x<0.5

Towards the larger numbers (right)

Solve: 2n+6<4

n<-1 ... Graph: open circle at -1, shade to the left

Solve: 5n-4>11

n>3... Graph: open circle at 3, shade to the right

consistent

dependent

elimination method

inconsistent

a system of equations with at least one solution. [p. 365]

a consistent system with infinite solutions. [p. 365]

a way to solve a system of linear equations by adding or subt…

a system of equations with no solutions. [p. 365]

consistent

a system of equations with at least one solution. [p. 365]

dependent

a consistent system with infinite solutions. [p. 365]

Take the converse and negate it

Take the conditional and negate it

When the exterior angle is equal to th…

Assume the opposite... Prove its wrong... Si…

Contrapositive

Inverse

Exterior angle inequality theorem

List the steps of an indirect proof

Take the converse and negate it

Contrapositive

Take the conditional and negate it

Inverse

composite function

index

inverse function

inverse relation

a function formed by evaluating one function with another fun…

the part of a radical that gives the degree of the root.

a function that "undoes" what the original function does.

a relationship in which one variable decreases when another v…

composite function

a function formed by evaluating one function with another fun…

index

the part of a radical that gives the degree of the root.

addition property of equality

equation

inequality

inverse operations

the property that states that if you add the same number to b…

an algebraic or numerical sentence that shows that two quanti…

a mathematical sentence that compares two expressions using t…

operations that undo each other; addition and subtraction are…

addition property of equality

the property that states that if you add the same number to b…

equation

an algebraic or numerical sentence that shows that two quanti…

Equation

Solution

Equivalent Equation

Subtraction Property of Equality

a mathematical sentence that contains an equals sign, stating…

a replacement value for the variable in an open sentence. A v…

two or more equations with the same solution

if you subtract the same number from each side of an equation…

Equation

a mathematical sentence that contains an equals sign, stating…

Solution

a replacement value for the variable in an open sentence. A v…

Linear System / System of Equations

Point of Intersection

Solving a system by GRAPHING

Solving a system by SUBSTITUTION

Two or more equations with the same variables 2x+ 3y = 2 ... 5x…

Always an ORDERED PAIR (x,y)

1. Put in slope-intercept form... 2. Graph BOTH... 3. Find where th…

Linear System / System of Equations

Two or more equations with the same variables 2x+ 3y = 2 ... 5x…

Point of Intersection

Always an ORDERED PAIR (x,y)

composite function

inverse function

nth root

principal root

a combination of two functions such that the output from the…

a relation formed by reversing x and y in a function. [p. 405]

for any real numbers A and B, and any positive integer N, if…

the positive square root of a number. [p. 361]

composite function

a combination of two functions such that the output from the…

inverse function

a relation formed by reversing x and y in a function. [p. 405]

expression

equation

Operation Symbols

inequality

NOT a number sentence. NO relation symbol

a number sentence that DOES contain an equal sign

X, - , + , /

a number sentence that DOES have a relation symbol

expression

NOT a number sentence. NO relation symbol

equation

a number sentence that DOES contain an equal sign

graph of a linear inequality in one va…

equivalent inequalities

compound inequality

linear inequality in x and y

The set of points on a number line that represents all soluti…

Inequalities that have the same solutions

Two or more inequalities that are connected by the words 'and…

An inequality that can be written in the form ax+b<c where a,…

graph of a linear inequality in one va…

The set of points on a number line that represents all soluti…

equivalent inequalities

Inequalities that have the same solutions

The Exterior angle inequality theorem

If one side of a triangle is longer th…

If one angle of a triangle is larger t…

The perpendicular segment from a point…

The measure of an exterior angle of a triangle is greater tha…

then the angle opposite the first side is larger than the ang…

then the side opposite the first side is larger than the angl…

is the shortest segment from that point to that line/plane

The Exterior angle inequality theorem

The measure of an exterior angle of a triangle is greater tha…

If one side of a triangle is longer th…

then the angle opposite the first side is larger than the ang…

consistent

dependent

elimination method

inconsistent

A system of equations that has at least one solution

A consistent system that has infinitely many solutions.

If a=b and c=d, then a+c=b+d and a-c=b-d. In this method, you…

A system of equations that has no solution.

consistent

A system of equations that has at least one solution

dependent

A consistent system that has infinitely many solutions.

equation

solution of an equation

linear equation in one variable

inverse operation

A written statement indicating the equality of two expression…

the value(s) of the variable that make the equation true

can be written in the form ax=b, where "a" is a coefficient a…

the operation that reverses the effect of another operation

equation

A written statement indicating the equality of two expression…

solution of an equation

the value(s) of the variable that make the equation true

inequality

solution of an inequality

solution set

graph of an inequality

a mathematical sentence that compares quantities

a value that makes the inequality true

the set of all solutions of an inequality

shows the solution set on a number line

inequality

a mathematical sentence that compares quantities

solution of an inequality

a value that makes the inequality true

The Exterior Angle Inequality Theorem

A statement and its contrapositive are

Step one of writing indirect proofs

Step two of writing indirect proofs

The measure of an exterior angle of a triangle is greater tha…

logically equivalent

Assume temporarily that the conclusion is not true.

Reason logically until you reach a contradiction of a known f…

The Exterior Angle Inequality Theorem

The measure of an exterior angle of a triangle is greater tha…

A statement and its contrapositive are

logically equivalent

Equation

Solution

Equivalent equations

Subtraction property of equality

A sentence that two quantities are equal

Value of a variable that makes a equation true

Same solution

Two sides of a equation remain equal when you subtract the sa…

Equation

A sentence that two quantities are equal

Solution

Value of a variable that makes a equation true

Solution

Equation

Equivalent equations

Subtraction property of equality

The value of a variable that make an equation true

A sentence stating that two quantities are equal

Same solution

States that the two sides of an equation remain equal when yo…

Solution

The value of a variable that make an equation true

Equation

A sentence stating that two quantities are equal