College Algebra Acosta/Karwowski

Chapter 2 Systems of equations

CHAPTER 2 - SECTION 1 A 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 xf(x)g(x)

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 2 nd 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

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 2 Solving 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-B 3 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 = x + 8y – 12 z = 43 2x + y = - 3 2y + 16z = x – 3y + 4z = 6

CHAPTER 2 – SECTION 3 Linear 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

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 4 Systems 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 < 3 Y ≥ 2x - 4

Inequality Systems - examples 3x + 2y < 8 3y – 2x < 15

Example x+ y > 2 x – y > 5

Inequality Systems - examples y < 2x -3 y> 4 – 2x x< 4