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 Equal at approximately 87 years old in 2064. 100 Years of Life 80 (84.11, 87.06) 60 20 40 60 80 100 120 Years since 1980

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.

Find the Ordered Pairs that Satisfy Both 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 Only one point satisfies both equations (-2,-3) is the solution set of the system 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 4 8 8 6x -6x + 3y = 7 – 6x 3y = -6x + 7 3 3 3 y = -2x + 7/3

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 (1.45,-.6)

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

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 1 2 3 4 y = 4x – 6 -6 -2 6 10 y = -6x + 14 14 8 -4 -10 Since (2,2) is a solution to both equations, it is a solution of the linear system.

3.2 Using Substitution 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

Example 1 y = x – 1 3x + 2y = 13 3x + 2(x – 1) = 13 3x + 2x – 2 = 13 5 5 x = 3 y = 3 – 1 y = 2 Solution set for the linear system is (3,2).

Example 2 2x – 6y = 4 3x – 7y = 8 2x – 6y +6y = 4 +6y 2x = 6y + 4 2 2 2 x = 3y + 2 3(3y + 2) – 7y = 8 9y + 6 – 7y = 8 2y + 6 -6 = 8 -6 2y = 2 2 2 y = 1 x = 3(1) + 2 x = 5 The solution set is (5, 1).

Using Elimination Adding left and right sides of equations If a=b and c=d, then a + c = b + d Substitute a for b and c for d a + c = a + c both sides are the same

Using Elimination Multiply both equations by a number so that the coefficients of one variable are equal in absolute value and opposite sign. Add the left and right sides of the equations to eliminate a variable. Solve the equation. Substitute the solution into one of the equations and solve.

Example 1 5x – 6y = 9 + 2x + 6y = 12 7x + 0 = 21 7x = 21 7 7 x = 3 7 7 x = 3 2(3) + 6y = 12 6 -6 + 6y = 12 -6 6y = 12 6 6 y = 2 Solution set for the system is (3,2)

Example 2 3x + 7y = 29 6x – 12y = 32 -2(3x + 7y) = -2(29) -26 -26 y = 1 3x + 7(1) = 29 3x + 7 -7 = 29 -7 3x = 22 3 3 x = 22/3 Solution (22/3, 1)

Example 3 6x + 15y = 42 + 35x – 15y = 40 2x + 5y = 14 41x + 0 = 82 41 41 x = 2 2(2) + 5y = 14 4 -4 + 5y = 14 -4 5y = 10 5 5 y = 2 2x + 5y = 14 7x – 3y = 8 3(2x + 5y) = 3(14) 6x + 15y = 42 5(7x – 3y) = 5(8) 35x – 15y = 40 Solution (2,2)

Using Elimination with Fractions 2x – 5y -1 3 9 3 3x – 2y 7 15 3 5 6x – 5y = -3 9( ) = ( )9 -2(6x – 5y) = (-3)-2 -12x + 10y = 6 15( ) = ( )15 3x – 10y = 21 +___________ -9x + 0 = 27 -9 -9 x = -3 6(-3) – 5y = -3 -18 +18 – 5y = -3 +18 -5y = 15 -5 -5 y = -3

Inconsistent Systems If the result of substitution or elimination of a linear system is a false statement, then the system is inconsistent. y = 3x + 3 -1(y) = -1(3x + 3) -y = -3x – 3 + y = 3x – 2 0 = 0 – 2 0 ≠ -2 False y = 3x + 3 y = 3x – 2 3x + 3 = 3x – 2 3x -3x + 3 = 3x -3x – 2 3 ≠ - 2 False

Dependent Systems If the result of applying substitution or elimination to a linear system is a true statement, then the system is dependent. y -3x = 3x -3x – 4 -3(-3x + y) = (-4)-3 9x + -3y = 12 + -9x + 3y = -12 0 = 0 True y = 3x – 4 -9x + 3y = -12 -9x + 3(3x – 4) = -12 -9x + 9x -12 = -12 -12 = -12 True

Solving systems in one variable To solve an equation A = B, in one variable, x, where A and B are expressions, Solve, graph or use a table of the system y = A y = B Where the x-coordinates of the solutions of the system are the solutions of the equation A = B

Systems in one variable y = 4x – 3 y = -x + 2 4x – 3 = -x + 2 4x +x – 3 = -x +x + 2 5x – 3 +3 = 2 +3 5x = 5 5 5 x = 1 (1,1)

Graphing to solve equations in one variable -2x + 6 = 5/4x – 3 (3,1) -2x + 6 = -4 (6,-4) (3,1) (6,-4)

Using Tables to Solve Equations in One Variable -2x + 7 = 4x – 5 Solution (2,3) x y -2 11 -13 -1 9 -9 7 -5 1 5 2 3 4

3.3 Systems to Model Data Predict when the life expectance of men and women will be the same. L = W(t) = .114t + 77.47 L = M(t) = .204t + 69.90 .114t -.114t + 77.47 = .204t + 69.90 -.114t 77.47 -69.90 = .09t + 69.90 -69.90 7.57 = .09t t ≈ 84.11

Solving to Make Predictions 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? A = W(t) = 13t + 4 A = M(t) = 6t + 84

Using substitution 13t + 4 = 6t + 84 13t + 4 -4 = 6t + 84 -4 7 7 t ≈ 11.43 Equal in 1961~1962