MATH 21
Lesson Eight

CHAPTER #10 - SYSTEMS OF EQUATIONS (Pages 469 to 531) - We have previously investigated the linear equation in two variables. We showed that it graphed as a straight line. In this chapter we are going to investigate systems of these equations.

1. A man invested a total of $4000 in securities.  Part of the money was invested at 5%, the rest at 6%.  His annual income from both investments was $230.  Find the amount of money he invested at each rate. 
   
2. A woman invested a total of $20,000 in three different common stocks, one paying a 6% annual dividend, one paying a 7% annual dividend, and the third paying an 8% annual dividend.  At the end of the first year, the sum of the dividends from the 6% and the 7% stocks was $940, and the sum of the dividends from the 6% and the 8% stocks was $720.  Find the amount of money she invested in each stock. 

LESSON #10.1 - The General Objectives of this lesson is to review systems of linear equations in two variables

PERFORMANCE OBJECTIVES - At the end of this lesson the student will be able to solve a system of two linear equations involving two variables using the substitution method.

10.1: Systems of Two Linear Equations in Two Variables - The general form of such a system is -

Eqn 1:   a₁₁x + a₁₂ y = c₁
Eqn 2:   a₂₁x + a₂₂ y = c₂

where the a_ij's and c_I's are real numbers. If we graph these two straight lines the following possibilities can occur, and as we can see, of these three cases only the consistent system has a unique solution. 

Figure 10:1 - Possible Graphs of Two Equations With Two Unknowns

Graphical Solution -  Consider the system -

x + 3y = 9
x - y = 1

Solving each of these equations for y, and then graphing them yields the following result.

Figure 10:2 - Graph of a Consistent System of Two Equations With Two Unknowns

 As you can see the two equations graph as straight lines, and intersect at the point P(3,2). This is the solution to the system of equations, and it can be checked by substituting x = 3 and y = 2 into both equations.

3 + 3(2) = 3 + 6 = 9 o.k.

3 - 2 = 1 o.k.

Substitution Method - The problem with graphical approaches is that it is difficult to get exact answers. The substitution method solves that problem. Consider our system again -

x + 3y = 9
x - y = 1

Step 1 - Use one of the equations to express one of the variables in terms of the other variable. (Either equation can be used, however, I recommend that you use the simplest one.) Using the second one yields -

x = y + 1

Step 2 - Substitute this result into the other equation.

y + 1 +3y =9

Step 3 - Solve for the remaining variable.

4y = 8

y = 2

Step 4 - Substitute this result into the equation in Step 1 to find the other variable.

x = 3

Check - Substitute the values of the variables into the original equations to check the answers.

Board and Class Problems - Pages 476 to 477/38, 44.

LESSON #10.2 - The General Objectives of this lesson is to expand our study of systems of linear equations in two variables

PERFORMANCE OBJECTIVES - At the end of this lesson the student will be able to solve systems of linear equations involving two variables using the elimination-by-addition method.

 10.2 - The Elimination-by-Addition Method - Although the substitution technique can be used to solve any system of equations there are other procedures that are sometimes easier to use. One of them is the elimination-by-addition method. This method involves the replacement of systems of equations with simpler equivalent systems until we obtain a system whereby we can easily extract the solutions. Equivalent systems of equations are systems that have exactly the same solution set..

Rules to Produce Equivalent Systems -

• Both sides of an equation can be multiplied by any nonzero real number.
• Any equation of a system can be replaced by the sum of that equation and a nonzero multiple of another equation.

Elimination Method - This technique is sometimes faster than the substitution method. If coefficients of one of the variables have the same magnitude, the addition or subtraction of the two equations yields one equation in one unknown.

 Consider the system -

x + 5y = -2
3x - 4y = -25

Step 1 - Multiply one or both equations by suitable integers, such that the magnitudes of the coefficients of one of the variables are the same.

Multiplying the first by 3 yields -

3x + 15y = -6
3x - 4y = -25

Step 2 - Subtract (or add) the second equation from the first.

19 y = 19

Step 3 - Solve for the remaining variable.

y = 1

Step 4 - Substitute this result into the equation in Step 1 to find the other variable.

x + 5y = -2
x + 5(1) = -2
x = -7

Check - Substitute the values of the variables into the original equations to check the answers.

-7 + 5*1 = -2 o.k.
3*(-7) - 4*1 = - 25 o.k.

 Which Method to Use - Both the substitution and elimination methods can be used to solve any system of linear equations in two variables. It is really an individual preference.

 Board and Class Problems - Pages 485 to 487/46.

LESSON #10.3 - The General Objectives of this lesson is to expand our study of systems of linear equations to include three variables.

PERFORMANCE OBJECTIVES - At the end of this lesson the student will be able to explain the geometric interpretation of a unique solution to a system of three linear equations involving three variables.

 10.3 - Systems of Three Linear Equations in Three Variables - The general form of such a system is -

a₁₁x + a₁₂y + a₁₃z = c₁
a₂₁x + a₂₂y + a₂₃z = c₂
a₃₁x + a₃₂y + a₃₃z = c₃

where the a's and c's are real valued coefficients. Geometrically, each one of these equations describes a plane in 3-dimensional space as shown for the following example.

x + y + z = 1

graphs as -

Figure 10:3 - Graph of the Plane      z = 1 - x - y.

If we had two equations such as -

x + y + z = 1
x + y -z = 0

 you would get two intersecting planes as shown in Figure 10:4.

 Figure 10:4 - Intersection of the Planes    x + y + z = 1  and    x + y - z = 0.

The intersection of these two planes forms a straight line in 3-dimensional space. If this line is intersected by a third plane expressed by another equation, there would be a unique point in space characterized by a triplet of the form -

(x,y,z)

common to all three planes. This would be the solution to the three equations.

There are a variety of techniques available to solve a system of three linear equations with three variables. One of the simplest to understand is called Cramer's Rule.

LESSON #10.5 - The General Objectives of this lesson is to learn Cramer's Rule as it is applied to a system of two linear equations in two variables

PERFORMANCE OBJECTIVES - At the end of this lesson the student will be able to solve systems of linear equations involving two variables using Cramer's Rule.

 10.5 - Determinants - Associated with any system of two linear equations involving two variables -

a₁₁x + a₁₂ y = c₁
a₂₁x + a₂₂ y = c₂

is the array of numbers

 which is called a square matrix. For each square matrix that has real number entries, there is a real number called the determinant of the matrix. The determinant is written as -

 and is defined by -

detA = a₁₁a₂₂ - a₁₂a₂₂

Associated with the linear system are two other matrices obtained by replacing the coefficients in each column by the constants c₁ and c₂ respectively, as shown as follows.

 Their determinants are detA_x and detA_y respectively.

Cramer's Rule - Given the system

a₁₁x + a₁₂ y = c₁
a₂₁x + a₂₂ y = c₂

then

Board and Class Problem - Using Cramer's Rule solve the following system for x and y.

x + 5y = -2
3x - 4y = -25

LESSON #10.6 - The General Objectives of this lesson is to extend Cramer's Rule to linear systems involving three equations and three unknowns.

PERFORMANCE OBJECTIVES - At the end of this lesson the student will be able to solve systems of linear equations involving three variables using Cramer's Rule.

 10.6 - 3 x 3 Determinants and Systems of Three Linear Equations in Three Variables - Cramer's Rule can be extended to any linear system, however, its application to systems involving more than three variables becomes cumbersome.

Cramer's Rule for Three Variables - Given the system

Explain how these determinants are evaluated by -

• products of diagonals.
• expansion by minors.

Board and Class Problems – Pages 512 to 514/2, 12.

LESSON #10.7 - The General Objectives of this lesson is to investigate systems of nonlinear equations.

PERFORMANCE OBJECTIVES - At the end of this lesson the student will be able to solve systems of nonlinear equations.

10.7 Systems Involving Nonlinear Equations and Systems of Inequalities - The first three techniques we used for solving linear systems can also be used when at least one of the equations involved is nonlinear.To illustrate the graphical method, and the substitution method, consider the following system.

x² + y² = 25

x + y = 5

The graph of these two equations is shown in Figure 10.5. 

 Figure 10.5 – The Graph of a Circle and a Straight Line

As is shown the straight line intersects the circle at two points.

P₁(0,5) and P₂(5,0)

These are the solutions to the system.

The third technique, or the elimination method is most appropriate when dealing with two equations where one of the variables is raised to the same power. Consider the system -

x ² + 2y ² = 8

x ² - y ² = 1

Board and Class Problems – Page 521/2, 12.

 HOMEWORK -  Lesson Eight Assignment

COMMENTS -

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