Chapter 5.1

Systems of Linear Equations

Linear Systems

The definition of a linear equation given in Chapter 1 can be extended to more variables; any equation of the form

(not all of which are 0) and b, is a linear equation or a first-degree equation in n unknowns.

bxaxaxa nn 2211

naaa ,, , numbers realfor 21

A set of equations is called a system of equations.

The solutions of a system of equations must satisfy every equation in the system. If all the equations in a system are linear, the system is a system of liner equations, or a linear system.

The solution set of a linear equation in two unknown (or variables) is an infinite set of ordered pairs.

Since the graph of such an equation is a straight line, there are three possibilities for the solution of a system of two linear equations in two unknowns, as shown in the figure.

Consistent Inconsistent Dependent

Substitution Method

In a system to two equations with two variables, the substitution method involves using one equation to find an expression for one variable in terms of the other, then substituting into the other equation of the system.

Solve the system.

1123 yx3 yx

Elimination Method

Another way to solve a system of two equations, called the elimination method, uses multiplication and addition to eliminate a variable from one equation. To eliminate a variable, the coefficients of that variable in the two equations must be additive inverses.

Elimination Method

To achieve this, we use properties of algebra to change the system to an equivalent system, one with the same solution set. The three transformations produce an equivalent system are listed here.

Elimination Method

Solve the system.

1 4 3 yx123 2 yx

Solve the system.

4 2 3 yx7 4 6 yx

Solve the system.

4- 2 8 yx2 4 yx

Applying Systems of Equations

Many applied problems involve more than one unknown quantity. Although some problems with two unknowns can be solved using just one variable, it is often easier to use two variables

Applying Systems of Equations

To solve a problem with two unknowns, we must write two equations that relate the unknown quantities. The system formed by the pair of equations can then be solved using the methods of this chapter.

Title IX legislation prohibits sex discrimination in sports programs. In 1997 the national average spent on two varsity athletes, one female and one male, was $6050 for Division I-A schools. However, average expenditures for a male athlete exceeded those for a female athlete by $3900. Determine how much was spent per varsity athlete for each gender.

In 1997 the national average spent on two varsity athletes, one female and one male, was $6050 for Division I-A schools.

60502

yx

x is the male expenditures. y is the female expenditures.

However, average expenditures for a male athlete exceeded those for a female athlete by $3900.

yx 6050

yx 6050

60502

yx 12100 yx

3900 yx

12100 yx3900 yx

16000 2 x8000 x

3900 y - 8000

y 3900- 8000 y 1004

Solving Linear Systems with Three Unknowns(Variables)

Earlier we saw that the graph of a linear equation in two unknowns is a straight line. The graph of a linear equation in three unknowns requires a three-dimensional coordinate system.

Solving Linear Systems with Three Unknowns(Variables)

The three number lines are placed at right angles. The graph of a linear equation in three unknowns is a plane.

Some possible intersections of planes representing three equations in three variables are shown.

To solve a linear system with three unknowns, first eliminate a variable from any two of the equations. then eliminate the same variable from a different pair of equations.

Eliminate a second variable using the resulting two equations in two variables to get an equation with just one variable whose vlaue you can now determine.

Find the values of the remaining variables by substitution. Solutions of the system are written as ordered pairs.

3693 zyx

Solve the system.

22 zyx

2 zyx

42 zyx

Solve the system.

943 zyx

Using Systems of Equations to Model Data

Applications with three unknowns usually require solving a system of three equations.

We can find the equation of a parabloa in the form

y = ax2 + bx + c

by solving a system of three equations with three variables.

Find the equation of the parabola y = ax2 + bx + cthat passes through the points (2,4), (-1,1), and (-2,5)

c2b2a4 2 c2b4a4or

c1-b1-a1 2 cba1or

c2-b2-a1 2 cb2a45or

An animal feed is made from three ingredients: corn, soybeans, and cottonseed. On unit of each ingredient provides units of protein, fat, and fiber as shown in the table.

How many units of each ingredient should be used to make a feed that contains 22 units of protein, 28 units of fat, and 18 units of fiber.

22 .2z .4y .25x 28 .3z .2y .4x

18 .1z .2y .3x