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
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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.
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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) = 0.114t + 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.
• Isolate a variable on one side of either equation
• Substitute the expression for the variable into the other equation
• Solve the second equation
• Substitute the solution into one of the equations
• In 1950, there were 4 women nursing students at a private college and 84 men. If the number of women nursing students increases by 13 a year and the number of male nursing students by 6 a year. What year will the number of male and female students be the same?