Systems of equations lesson 5

MATH10 ALGEBRA

SYSTEMS OF EQUATIONS

Systems of Equations(Algebra and Trigonometry, Young 2nd Edition, page 874-904)

GENERAL OBJECTIVE

At the end of the chapter the students are expected to:

Week 6 Day 1

• Solve systems of equations in two variables with the substitution method and the elimination method.

• Graph systems of linear equations.• Understand that systems of linear equations may have one

solution, no solution, or infinitely many solutions.• Solve systems of equations in three variables employing

combination of the elimination and substitution methods.

SYSTEMS OF LINEAR EQUATIONS IN TWO VARIABLES

TODAY’S OBJECTIVE

• To solve systems of linear equations in two variables using the substitution method.

• To solve systems of linear equations in two variables using the elimination method.

• To solve systems of linear equations in two variables by graphing.• Understand that a system of linear equations has either one

solution, no solution or infinitely many solution.

At the end of the lesson the students are expected to:

Week 6 Day 1

DEFINITION

SYSTEMS OF EQUATIONS

A system of equations is a set of equations that involve the same variables. To solve a system of equations means to find the solution that satisfies both equations.

Example:

7y4x5yx2

11yx36y2x

Week 6 Day 1

Solutions are given as an ordered pair of the form (x,y) .

ALGEBRAIC GRAPHICALSOLUTION 1y and 4x 4,1

61246y2x

1 Equation

1114311yx3

2 Equation

equations both satisfy 1y and 4x

-4

6

lines both on lie 4,1 point The

Check

Example:

INTERPRETATION

11yx36y2x

y

x

x+2y=6 3x-y

=11

Week 6 Day 1

THREE TYPES OF SOLUTIONS TO SYSTEMS OF LINEAR EQUATIONS

SPECIAL NAME NUMBER OF SOLUTIONS GRAPHICAL INTERPRETATION

-4

6

1. Independent System

One solution

Lines have different slopes.

y

x

Week 6 Day 1



-4

6

2. Inconsistent System

No solution

Lines are parallel (same slopes and different y - intercepts.)

y

x

Week 6 Day 1



-4

6

3. Dependent System

Infinitely many solution

Lines coincide (same slopes and same y – intercepts).

y

x

Week 6 Day 1

Three methods of solving systems of linear equations:

Algebraic Methods – used to find exact solutions. a. substitution methodb. elimination method

Graphing Method – typically used to give a visual interpretation and confirmation of the solution.

METHODS OF SOLVING SYSTEMS OF LINEAR EQUATIONS

Week 6 Day 1

SUBSTITUTION METHOD

STEPS:

1. Solve one of the equations for one variable in terms of the other variable.

2. Substitute the expression found in step 1 into the other equation. The result is an equation in one variable.

3. Solve the equation obtained in step 2.4. Back substitute the value found in step 3 onto the expression

found in step 1.5. Check that the solution satisfies both equations.

Week 6 Day 1

DETERMINING BY SUBSTITUTION THAT A SYSTEM HAS ONE SOLUTION

878 page 1.1.9.Ex Classroom

9y2x34yx

.1

878 page*1.1.9.Ex Classroom

1yx1ayx

solution. one has system following the that such aof values the Determine .2

Week 6 Day 1

DETERMINING BY SUBSTITUTION THAT A SYSTEM HAS NO SOLUTION


0y15x61y5x2

.1


9y6x33ay2x11


Week 6 Day 1

DETERMINING BY SUBSTITUTION THAT A SYSTEM HAS INFINITE SOLUTION


6y8x23xy4

.1


x105ay 1y5x2


Week 6 Day 1

ELIMINATION METHOD

STEPS:

1. Multiply the coefficients of one or both of the equations so that one of the variables will be eliminated when two equations are added.

2. Eliminate one of the variables by adding the expression found in Step 1 to the other equation. The result is an equation in one variable.

3. Solve the equation obtained in Step 2.4. Back substitute the value found in Step 3 into either of the

original equation.5. Check that the solution satisfies both equations.

Week 6 Day 1

DETERMINING BY ELIMINATION THAT A SYSTEM HAS ONE SOLUTION

880 page 4# Example

11yx45yx2

.1

880 page*4.1.9.Ex Classroom umbern real any is a where

1yx 1ayx

neliminatio using Solve .2

881 page 5.1.9.xClassroomE

9y2x34yx

.3

881 page 6# Example

9y7x51y2x3

.4

Week 6 Day 1

DETERMINING BY ELIMINATION THAT A SYSTEM HAS NO SOLUTION

882 page 4# 9.1.7 .Ex Classroom

0y15x61y5x2

inationlime gsinu Solve. 1

882 page 7# Example

4y22x 7yx

neliminatio using Solve .2

Week 6 Day 1

DETERMINING BY ELIMINATION THAT A SYSTEM HAS INFINITELY MANY SOLUTION

882 page 8# Example

4y2x142yx7

.1

883 page (b) YourTurn

20y5010x- 2y5x

method neliminatio the Apply .2

Week 6 Day 1

GRAPHING METHOD

STEPS:

1. Write the equations in the slope-intercept form.2. Graph the lines.3. Identify the points of intersection.4. Check that the solution satisfies both equations.

1. Solve for the x and y intercepts.2. Graph the lines.3. Identify the points of intersection.4. Check that the solution satisfies both equations.

OR

Week 6 Day 1

DETERMINING BY GRAPH THAT A SYSTEM HAS ONE SOLUTION, NO SOLUTION OR INFINITELY MANY SOLUTION

884 page 9.1.9 .Ex Classroom

9y2x34yx

ygraphicall Solve .1

885 page (a) YourTurn

20y42x 1y2x

system given the solve to graphing Utilize .2

Week 6 Day 1

IDENTIFYING WHICH METHOD TO USE

Given any system of linear equations in two variables, any of the three methods ( substitution, elimination, or graphing) can be utilized. Elimination is preferred if it is easy to eliminate a variable by adding multiples of two equations. Use substitution if there is no obvious elimination.Use graphing to confirm the solution(s) found using either elimination or substitution.

18y 5x 1 20y-7x

c. 4y-2x1-2yx

b. 2yx-12y-x

a.

State which of the two algebraic methods (elimination or substitution) would be the preferred method to solve each system of linear equations.

EXAMPLE

Week 6 Day 1

Three methods of solving systems of linear equations:

Algebraic Methods – used to find exact solutions. a. substitution methodb. elimination method

Graphing Method – typically used to give a visual interpretation and confirmation of the solution.

SUMMARY

Three types of solutions to systems of linear equations:

One solution – the system is called an independent system - the lines formed are intersecting lines No solution - the system is called inconsistent system - the lines formed are parallel lines Infinitely many solutions –the system is called dependent system - the lines formed coincide.

Week 6 Day 1

SYSTEMS OF LINEAR EQUATIONS IN THREE VARIABLES

Week 6 Day 2

Week 6 Day 2TODAY’S OBJECTIVE

• To understand that a graph of linear equation in three variables correspond to a plane.

• To identify three types of solutions: one solution, no solution or infinitely many solutions.

• To solve systems of linear equations in three variables using the combination of both elimination method and the substitution method.


Week 6 Day 2DEFINITION

SYSTEMS OF EQUATIONS IN THREE VARIABLES

A linear equation in three variables x, y, and z is given byAx +By +C = D

where A, B, C, and D re real numbers that are not all equal to zero.All three variables have degree equal to one, which is why this is called equation in three variables .The graph of any equation in three variables requires three dimensional coordinate system.In two variables, the graph of a linear equation is a line, while in three variables the graph of a linear equation is a plane which can be thought of as an infinite sheet of paper.Solutions are given as an ordered pair of the form (x,y,z)

Week 6 Day 2

THREE TYPES OF SOLUTIONS TO SYSTEMS OF LINEAR EQUATIONS IN THREE VARIABLES

1. Independent - one solution2. Dependent - infinitely many solutions3. Inconsistent - no solution

Week 6 Day 2

Solution

One Solution

Week 6 Day 2

or

No Solution

Week 6 Day 2

Solution(line of intersection)

Infinitely Many Solutions

SOLVING SYSTEMS OF LINEAR EQUATIONS IN THREE VARIABLES USING ELIMINATION AND SUBSTITUTION

1. Reduce the system of three equations in three variables to two equations in two (of the same) variables by applying elimination.2. Solve the resulting system of two linear equations in two variables by applying elimination or substitution.3. Substitute the solution in Step 2 into any of the original equations and solve for the third variable.4. Check that the solution satisfies all three original equations.

STEPS

Week 6 Day 2

EXAMPLE

894 page 9.2.1 .Ex Classroom

-5 2z-3y-5x3 z3yx

8z5y3x4

:system the Solve .1

896 page YourTurn

2 z y x 1z 2x

0zyx

:system given the Solve .2

896 page 3 Example

6 zy2 x3

2 y x 4zy x2


897 page 4# Example

5 z2 y4 x2-1 z2 y 2x 3 z y2x


Week 6 Day 2

SOLVING SYSTEMS OF TWO LINEAR EQUATIONS IN THREE VARIABLES

Two linear equations in three variables will always correspond to two planes in three dimensions. There are two possibilities

1. No solution if the two planes are parallel2. Infinitely many solutions if the two planes intersect in a line.

Week 6 Day 2

y

x

zNo solution if the two planes are parallel

Week 6 Day 2

z

y

x

Infinitely many solutions if the two planes intersect in a line.Week 6 Day 2

solution

EXAMPLE

898 page 5# Example

2 z2y x7z yx

:system the Solve .1


0 z yb 1y b-x2


Week 6 Day 2

Week 6 Day 2SUMMARY

Graphs of linear equations in two variables are lines, whereas graphs of linear equations in three variables are planes.

Systems of linear equations in three variables have one of the three outcomes:

1. One solution (point)2. No solution (no intersection of all three planes)3. Infinitely many solutions (planes intersect along a line)

When the solution to a system of three linear equations is a line in three dimensions, we use parametric representations to express the solution.

Week 6 Day 3TODAY’S OBJECTIVE

• To solve systems of nonlinear equations.• To solve application problems involving systems of equations in

two and in three variables.


QUADRATIC SYSTEMS IN TWO VARIABLES

The most general form of a quadratic equation in the variables x and y is

R B...FA, where 0EyDxCyBxyAx 22

equation quadratic a have to present be must term xy the or y,x least at but however, present, be may terms the all Not 22

The graphs of these equations are circles and the conic sections which are to be discussed in analytic geometry.

Week 6 Day 3


1. One Linear, One Quadratic

1y5x203y15x8y2xy9x4

:system the Solve22

2. Two Quadratics

5y12x8

11y4 x5

:system the Solve

22

22

3.Two Quadratics, all terms containing the variable are of second degree

4y2xy3x

7y xy5x

:system the Solve

22

22

College Algebra Revised edition, Catalina D. Mijares page 241-250

Week 6 Day 3


4. Symmetric Quadratic Equation

2yxxy4yx

:system the Solve22

5.Other types which does not fall on the previous types.

015y6x2yx

035y4x18yx

:system the Solve

22

22

College Algebra Revised edition, Catalina D. Mijares page 241-250

Week 6 Day 3

APPLICATION INVOLVING SYSTEMS OF LINEAR EQUATIONS

Week 6 Day 3 Application Involving Systems of Linear Equations (Algebra and Trigonometry, Young 2nd Edition, page 886-891 and 899-904).

Week 6 Day 3

Start

Read and analyze the problem

Make a diagram or sketch if possible

Determine the unknown quantity.

Did you set up the equation?

Set up an equation, assign variables to represent what you are asked to find.

Ano yes

A

Solve the equation

Check the solution

Is the unknown solved?

no

yes

End

RECALLWeek 6 Day 3

Week 6 Day 3APPLICATION

1. Upon graduation with a degree of management information systems(MIS), you decide to work for a company that buys data from the states’ department of motor vehicles and sells to banks and car dealerships customized reports detailing how many cars at each dealership are financed through particular banks. Autocount Corporation offers you a $15,000 base salary and a 10% commission on your total annual sales. Polk Corporation offers you a base salary of $30,000 plus a 5% commission on your total annual sales. How many total sales would you have to make per year to make more money at Autocount? (# 59 page 890)

Week 6 Day 3

APPLICATION

2. A mechanic has 340 gallons of gasoline and 10 gallons of oil to make gas/oil mixtures. He wants one mixture to be 4% oil and the other mixture to be 2.5% oil. If he wants to use all of the gas and oil, how many gallons of gas and oil are in each of the resulting mixtures? (# 58 page 890)

3. A direct flight on Delta Airlines from Atlanta to Paris is 4000 miles and takes approximately 8 hours going East (Atlanta to Paris) and 10 hours going West ( Paris to Atlanta). Although the plane averages the same airspeed, there is a headwind while traveling west and a tailwind while travelling east resulting in different airspeeds. What is the average airspeed of the plane and what is the average wind speed ? (# 63 page 890)

Week 6 Day 3

APPLICATION

Suppose you’re going to eat only Subway sandwiches for a week (7 days) for lunch and dinner (total o0f 14 meals).

Sandwich Calories Fat (grams)

Mediterranean Chicken 350 18

Six Inch Tuna Roast Beef 430 19

Six In 290 5

www.subway.com

4. Your goal is a total of 4840 calories and 190 grams of fat. How many of each sandwich would you eat that week to obtain this goal? ( #33 page 901)

Week 6 Day 3

APPLICATION

5. Tara and Lamar decide to place $20,000 of their savings into investments. They put some in a money market account earning 3% interest, some in a mutual fund that has been averaging 7% a year, and some in a stock that rose 10% last year. If they put $6,000 more in the money market than in the mutual fund and the mutual fund and stocks have the same growth in the next year as they did in the previous year , they will earn $1,180 in a year. How much money did they put in each of the three investments? (# 39 page 902)

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Systems of equations lesson 5

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