3.1 Systems of Linear Equations

• Using graphs and tables to solve systems

• Using substitution and elimination to solve systems

• Using systems to model data

• Value, interests, and mixture problems

• Using linear inequalities in one variable to make predictions

Using Two Models to Make a Prediction

• When will the life expectancy of men and women be equal?– L = W(t) = 0.144t + 77.47– L = M(t) = 0.204t + 69.90

Years since 1980

Yea

rs o

f Li

fe

60

80

100

20 40 60 80 100 120

(84.11, 87.06)

Equal at approximately 87 years old in 2064.

System of Linear Equations in Two Variables (Linear System)

• Two or more linear equations containing two variables

y = 3x + 3

y = -x – 5

Solution of a System

• An ordered pair (a,b) is a solution of a linear system if it satisfies both equations.

• The solution sets of a system is the set of all solutions for that system.

• To solve a system is to find its solution set.

• The solution set can be found by finding the intersection of the graphs of the two equations.

• Graph both equations on the same coordinate plane– y = 3x + 3– y = -x – 5

• Verify– (-3) = 3(-2) + 3– -3 = -6 + 3– -3 = -3

– (-3) = -(-2) – 5 – -3 = 2 – 5 – -3 = -3

• Only one point satisfiesboth equations

• (-2,-3) is the solution set of the system

Find the Ordered Pairs that Satisfy Both Equations

Solutions for

y = 3x + 3

Solutions for

y = -x – 5

Solution for both (-2,-3)

Example• ¾x + ⅜y = ⅞• y = 3x – 5 • Solve first equation for y

– ¾x + ⅜y = ⅞– 8(¾x + ⅜y) = 8(⅞)– 24x + 24y = 56

4 8 8

– 6x -6x + 3y = 7 – 6x– 3y = -6x + 7

3 3 3

– y = -2x + 7/3

(1.45,-.6)

y = 3x – 5

-.6 = 3(1.45) – 5

-.6 = 4.35 – 5

-.6 ≈ -.65

y = -2x + 7/3

-.6 = -2(1.45) + 7/3

-.6 = -2.9 + 7/3

-.6 ≈ -.57

Inconsistent System

• A linear system whose solution set is empty– Example…Parallel lines never intersect

• no ordered pairs satisfy both systems

Dependent System

• A linear system that has an infinite number of solutions– Example….Two equations of the same line

• All solutions satisfy both lines

y = 2x – 2

-2x + y = -2• -2x +2x + y = -2 +2x• y = 2x – 2

One Solution System

• There is exactly one ordered pair that satisfies the linear system– Example…Two lines

that intersect in only

one point

Solving Systems with Tables

x 0 1 2 3 4

y = 4x – 6 -6 -2 2 6 10

y = -6x + 14 14 8 2 -4 -10

• Since (2,2) is a solution to both equations, it is a solution of the linear system.