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Intermediate Algebra Chapter 4 • Systems • of • Linear Equations Objective • Determine if an ordered pair is a solution for a system of equations. System of Equations • Two or more equations considered simultaneously form a system of equations. a1 x b1 y c1 a2 x b2 y c2 Checking a solution to a system of equations • 1. Replace each variable in each equation with its corresponding value. • 2. Verify that each equation is true. Graphing Procedure • 1. Graph both equations in the same coordinate system. • 2. Determine the point of intersection of the two graphs. • 3. This point represents the estimated solution of the system of equations. Graphing observations • Solution is an estimate • Lines appearing parallel have to be checked algebraically. • Lines appearing to be the same have to be checked algebraically. Classifying Systems • Meet in Point – Consistent – independent • Parallel – Inconsistent – Independent • Same – Consistent - Dependent Def: Dependent Equations • Equations with identical graphs Independent Equations • Equations with different graphs. Algebraic Check • Same Line a1 x b1 y c1 a2 x b2 y c2 a1 b1 c1 a2 b2 c2 Algebraic Check • Parallel Lines a1 x b1 y c1 a2 x b2 y c2 a1 b1 c1 a2 b2 c2 Algebraic Check • Meet in a point {(x,y)} a1 x b1 y c1 a2 x b2 y c2 a1 b1 c1 a2 b2 c2 Calculator Method for Systems • Solve each equation for y • Input each equation into Y= • Graph • Set Window • Use CalIntersect Calculator Problem y 2 x 3 x 2y 4 2, 1 Calculator Problem 2 3x 4 y 8 3 y x3 4 Calculator Problem 3 6x 2 y 4 y 3x 2 x, y | y 3x 2 Objective • Solve a System of Equations using the Substitution Method. Substitution Method • 1. Solve one equation for one variable • 2. In other equation, substitute the expression found in step 1 for that variable. • 3. Solve this new equation (1 variable) • 4. Substitute solution in either original equation • 5. Check solution in original equation. Althea Gibson – tennis player • “No matter what accomplishments you make, someone helped you.” Intermediate Algebra •The •Elimination •Method Notes on elimination method • Sometimes called addition method • Goal is to eliminate on of the variables in a system of equations by adding the two equations, with the result being a linear equation in one variable. • 1.Write both equations in ax + by = c form • 2. If necessary, multiply one or both of the equations by appropriate numbers so that the coefficients of one of the variables are opposites. Procedure for addition method cont. • 3. Add the equations to eliminate a variable. • 4. Solve the resulting equation • 5. Substitute that value in either of the original equations and solve for the other variable. • 6. Check the solution. Procedure for addition method cont. • Solution could be ordered pair. • If a false statement results i.e. 1 = 0, then lines are parallel and solution set is empty set. (inconsistent) • If a true statement results i.e. 0 = 0, then lines are same and solution set is the line itself. (dependent) Practice Problem x y 6 2 x 5 y 16 • Answer {(2,4)} Practice Problem Hint: eliminate x first 4 x 3 y 2 6 x 7 y 7 • Answer {(-7/2,-4)} Practice Problem 2x y 1 2 x y 3 Practice problem 3x 4 y 5 9 x 12 y 15 Special Note on Addition Method • Having solved for one variable, one can eliminate the other variable rather than substitute. • Useful with fractions as answers. Practice Problem – eliminate one variable and than the other 3 3 15 x y 4 4 4 4 5 x y3 3 3 • Answer: {(8/3,1/3)} Confucius • “It is better to light one small candle than to curse the darkness.” Intermediate Algebra 4.2 • Systems • Of • Equations • In • Three Variables Objective • To use algebraic methods to solve linear equations in three variables. Def: linear equation in 3 variables • is any equation that can be written in the standard form ax + by +cz =d where a,b,c,d are real numbers and a,b,c are not all zero. Def: Solution of linear equation in three variables • is an ordered triple (x,y,z) of numbers that satisfies the equation. Procedure for 3 equations, 3 unknowns • 1. Write each equation in the form ax +by +cz=d • Check each equation is written correctly. • Write so each term is in line with a corresponding term • Number each equation Procedure continued: • 2. Eliminate one variable from one pair of equations using the elimination method. • 3. Eliminate the same variable from another pair of equations. • Number these equations Procedure continued • 4. Use the two new equations to eliminate a variable and solve the system. • 5. Obtain third variable by back substitution in one of original equations Procedure continued • Check the ordered triple in all three of the original equations. Sample problem 3 equations (1) x y z 2 (2) 2 x y 2 z 1 (3) 3x 2 y z 1 Answer to 3 eqs-3unknowns •{(-2,3,1)} Bertrand Russell – mathematician (1872-1970) • “Mathematics takes us still further from what is human, into the region of absolute necessity, to which not only the actual world, but every possible world, must conform.” Cramer’s Rule • Objective: Evaluate determinants of 2 x 2 matrices • Objective: Solve systems of equations using Cramer’s Rule Determinant a b If A then det[ A] c d a b ad bc c d Cramer’s rule intuitive • Each denominator, D is the determinant of a matrix containing only the coefficients in the system. To find D with respect to x, we replace the column of s-coefficients in the coefficient matrix with the constants form the system. To find D with respect to y, replace the column of y- coefficients in the coefficient matrix sit the constant terms. Sample Problem: Evaluate: 3 2 2 4 • Answer = 16 Sample Cramer’s Rule problem • Solve by Cramer’s Rule 2 x 3 y 5 3x y 9 Cramer’s Rule Answer D 11 Dx 22 Dy 33 Dx 22 x 2 D 11 Dy 33 y 3 D 11 Senecca • “It is not because things are difficult that we do not dare, it is because we do not dare that they are difficult.” Intermediate Algebra 5.5 • Applications • Objective: Solve application problems using 2 x 2 and 3 x 3 systems. Mixture Problems • ****Use table or chart • Include all units • Look back to test reasonableness of answer. Sample Problem • How many milliliters of a 10% HCl solution and 30% HCl solution must be mixed together to make 200 milliliters of 15% HCl solution? Mixture problem equations x y 200 0.10 x 0.30 y 30 Mixture problem answers • 150 mill of 10% sol • 50 mill of 30% sol • Gives 200 mill of 15% sol Distance Problems • Include Chart and/or picture • Note distance, rate, and time in chart • D = RT and T = D/R and R=D/T • Include units • Check reasonableness of answer. Sample Problem • To gain strength, a rowing crew practices in a stream with a fairly quick current. When rowing against the stream, the team takes 15 minutes to row 1 mile, whereas with the stream, they row the same mile in 6 minutes. Find the team’s speed in miles per hour in still water and how much the current changes its speed. Distance problem equations 0.25( x y ) 1 0.1( x y ) 1 x y 4 x y 10 Answer • Team row 7 miles per hour in still water • Current changes speed by 3 miles per hour Joe Paterno – college football coach • “The will to win is important but the will to prepare is vital.”