Systems of Two Equations in Two Unknowns• In order to explain what we mean by a system of equations, we

consider the following: Problem. A pile of 9 coins consists of nickels and quarters. If the total value of the coins is $1.25, how many of each type of coin are there?

Solution. Let x = the number of nickels and let y = the number of quarters. The requirements of the problem can be expressed as:

This is an example of a system of two equations in two unknowns, and we want a pair of numbers (in this case x = 5, y = 4) which satisfy both equations. Such a pair of numbers is called a solution of the system.

.125255

9

yx

yx

Solution by Substitution

• Suppose we have a system of two equations in two unknowns. If we can use one of the equations to express one variable in terms of the other variable, then we can substitute this expression into the second equation, which results in a single equation containing a single unknown.

• For the system from the previous slide, we solve by substitution:

.49 ,5

10020

12522520

125)9(255

9

xyx

x

x

xx

xy

Another Example for Solution by Substitution

• Solve the system of equations:

• Solution.

• Note: In this example, the system has two solutions.

.1

2522

yx

yx

4y 3

3or 4

0)3)(4(

012

25)1(

1

2

22

y

xx

xx

xx

xx

xy

Solution by Graphing

• If we graph the two equations of a system, then the coordinates of the points of intersection of the two graphs must be the solutions of the system.

• Example from previous slide:

(4,3)

(3,4)

2522 yx

1 yx

Consistent and Inconsistent Systems

• A consistent system has one or more solutions. We have seen examples of consistent systems in previous slides.

• An inconsistent system has no solutions. Geometrically, this happens when the two graphs have no points of intersection.

• Example. The system given below is inconsistent. Do you see why?

.322

1

yx

yx

Solution by Elimination

• The method of elimination seeks to combine the equations of a system in such a way to eliminate one of the unknowns.

• Example. Solve the system:

Now, substitute the values for x into one of the original equations and solve for y. This results in three solutions: (5, 0), (4, 3), (4, 3).

5

252

22

yx

yx

4or 5

0)4)(5(

020

1st. fromequation 2ndSubtract 20 2

2

xx

xx

xx

xx

Systems of Linear Equations

• A system consisting only of equations that are of first degree in x and y is called a system of linear equations or simply a linear system. When we graph a system of two linear equations, we are graphing two straight lines. There are 3 possibilities: 1. The two lines intersect at a point. The system is consistent and has a unique solution—the point of intersection.

2. The two equations are different forms of the same line. The system is consistent and has an infinite number of solutions—all points on the line.

3. The two lines are parallel. Since the lines do not intersect, the system is inconsistent and has no solution.

Solving an Inconsistent System by Elimination

• Problem.

• Solution.

.322

1

yx

yx

.3 22

2.by equation first Multiply 22 2

yx

yx

equations. theAdd .10

Since assuming that a solution exists leads to a contradiction (0 = 1), we conclude that no solution exists.

Summary of Systems of Eqns in Two Unknowns; We discussed

• Solution of a system as a pair of numbers

• Solution by substitution

• Solution by graphing

• Consistent and inconsistent systems

• Solution by elimination

• Systems of two linear equations and their solutions