College Algebra Acosta/Karwowski. Chapter 2 Systems of equations.

College Algebra

Acosta/Karwowski

Chapter 2

Systems of equations

CHAPTER 2 - SECTION 1A graphical explanation of systems

Definitions

• a system is a set of 2 or more equations that are statements about the same domain and range.

• the solution of a system is the point or points at which both statements are true - this is the mathematical definition of intersection.

• Therefore the solution of a system is where the graphs of the 2 equations intersect.

• If the equations are linear this can happen once, never, or everywhere

What the graphs tell us

• consistent and inconsistent• independent and dependent

f(x)g(x)

k(x)

m(x)

j(x)

n(x)

Using graphs to solve a system

• Given a graph the solution is the point of intersection

Using systems to solve linear equations in one variable

• An equation in one variable can be viewed as a system

• Ex: 3(x – 5) + 2x = 7x – 9 becomes f(x) = 3(x – 5) + 2x and g(x) = 7x – 9 When the solution on the screen appears the value for x is the solution to the equation•

Solving numerically

• Before technology this was called guess and check

• now it is called using a table to solve• given 2 equations you generate a table with 3

columns when you graph. The Solution is the place where column 2 is the same as column 3

Examples

• Find the solution for the following system

x f(x) g(x)

-5 12 7.2

-4 11.3 7.7

-3 10.6 8.2

-2 9.9 8.7

-1 9.2 9.2

0 8.5 9.7

Using your graphing calculator• Solve equations for y • Enter the equations in calculator• Press graph - adjust window so the intersection shows (for linear

equations this isn’t really necessary)• Press 2nd button and trace button – choose number 5 in the menu.• a cursor appears on one of the lines – use arrows to move the cursor close

to the point of intersection (with linear equations this is not necessary but it IS necessary when we do non- linear systems)

• the calculator is displaying the equation for the line at the top and a question at the bottom. Press enter to answer yes - another question appears – press enter again- ANOTHER question appears – press enter AGAIN. (in other words press enter 3 times)

• now there are 2 numbers displayed at the bottom of the screen. This is the point of intersection

Using technology

• Solve both equations for y• Enter the equations• Press 2nd button and graph button and a table

appears. Use the up and down arrows to scroll through the table or go to table settings and choose a different start number

• if you can’t find a match you have to go to table set and choose a new

• This is EXTREMELY tedious -

Major draw back for this method

• the calculator will give you decimal values and if they are repeating decimals or irrational numbers this may not be accurate enough.

CHAPTER 2 – SECTION 2Solving 2 variable systems

Underlying theory

• If there are 2 variables in the problem I have infinite solutions

• If I can eliminate one of the variables I get an equation with one solution

• The second equation gives me a way to eliminate one of the variables

• There are 2 pieces of mathematics that accomplish this One is called substitution The other is called elimination or sometimes addition and multiplication method

Substitution

• Substitution is replacing one “number” with another form of the number that is equal to it

• You do this when you evaluate a function Ex: find 3x = y when x = -5• You can do this with full equations also

Examples

• y = 3x + 2 3y – 2x = 8

• 3x – y = 26 3y + 2x = -1

• 2x + 3y = 12 6y = 15 – 4x

• f(x) = 3x + 8 g(x) = 7 – 2x

• 2x – 3y = 12 3x + 6y = 32

Elimination

• Sometimes simply called the addition/multiplication method because that is what you do

• For any equation you can add anything you want as long as you add to both sides

• This includes adding equations - as long as you add both sides

• This is helpful if one of the variable terms match - If they don’t you have to multiply to make them match

examples

• 2x + 4y = 26 -2x – 6y = -32

• x – y = 14 x + 4y = 10

• 2x + 3y = 19 5x – 4y = -10

• 3x = -5 – 7y 2y + 6x = 14

• 2x + 4y = 3 8y – x = 11

• 2(x – 8) + 3y = -18 y – 3x = 25

More examples

•

CHAPTER 2-SECTION 2-B3 variable systems

Systems in 3 variables

• The system requires 3 equations in order to solve.• Geometrically each equation defines a plane in

space and the solution is a single point where all three planes intersect.

• The intersection of any 2 planes is a line.• The 3 planes can intersect in such a way that the

intersection is a line (dependent system) or they can meet only 2 at any given spot (or be parallel) (inconsistent system)

Solving symbolically/analytically

• Use one equation to remove one variable from the other 2 equations

• Solve the resulting two variable system• Return and find the value of the third variable

• To remove the first variable the techniques are the same as for two variable systems.

Example: by substitution• x + y + z = 7 (eq1) 2x – 3y + z = -62 (eq2) 2x + 4y + 3z = 42 (eq3)

• Solution (-5,16,-4)

Example: solve by elimination• 2x – 3y + 5z = -8 (eq1) • 3x + 3y – 2z = 9 (eq2)• 5x – y + z = 8 (eq3)

• x = 2 y = -1 z=-3

Inconsistent: analytically

• If at any time all variables cancel and the equation is false the system is inconsistent

• Example: • 5x + 9y + z = 9 3x + 4y + z = 12 4x + 3y +2z = 12

Dependent - analytically

• If at any time the variables disappear and the resulting equation is true you have a dependent system

• Example:• 6x – 4y + z = 9 8x – 3y + z = 7 4x – 5y + z = 11

More Examples

• 2x – 3y +2z = -7 x + 4y – z = 10 3x + 2y + z = 4

• 2x + 5y – z = 9 x - y + 2z = -1 x + 3y - z = 6

• 2x + y + 3z = 2 x - 11y + 4z = -54 -5x + 8y – 12 z = 43

• 2x + y = - 3 2y + 16z = -18 -7x – 3y + 4z = 6

CHAPTER 2 – SECTION 3Linear inequalities

2 variable inequalities

• The solution set for a one variable inequality is an interval - it has a start and a stop and can be described easily

• The solution set of a two variable inequality is a section of the plane which can only be described in the most abstract sense therefore graphs on linear inequalities are very helpful

Graphing Linear inequalities

• Graph line.• Determine the direction of shading either by

analyzing the equation or substituting a point• Shade the section that is true for the

inequality [check a point to verify]• Most graphing utilities have a way to do this

on them

Example

• y > 2x – 7

Example

• 2x + 5y < 10

Example

• 2

7

3 xy

Solutions/test point

• a given point is a solution to the problem if it lies in the solution area

• a given point is a solution if it makes the algebraic sentence true

Example:

• Mark need to buy notebooks and pencils. The notebooks cost $4.50 each and the pencils packages cost $2.75 each. He can spend no more than $50.

• write an algebraic inequality to model this.• graph the inequality• can he buy 7 notebooks and 10 packages of

pencils

CHAPTER 2 – SECTION 4Systems of linear inequalities

Systems of inequalities

• A problem that involves two or more conditions on the domain and range is a system

• Like single inequalities (only worse) a graph is the best way to see the solution set for this type of problem

• the solution set is the intersection of the two inequalities – if it exists it is one section of the plane

Systems of inequalities

• Graph both lines • Determine the section which makes BOTH

inequalities true and shade ONLY that section• You can use test points or analyze the

statement

Inequality Systems - examples

Y < 3Y ≥ 2x - 4

Inequality Systems - examples

3x + 2y < 83y – 2x < 15

Example

• x+ y > 2 x – y > 5

Inequality Systems - examples

y < 2x -3y> 4 – 2xx< 4