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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 . Checking a solution to a system of equations.
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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.
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
Algebraic Check • Parallel Lines
Algebraic Check • Meet in a point {(x,y)}
Calculator Method for Systems • Solve each equation for y • Input each equation into Y= • Graph • Set Window • Use CalIntersect
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 emptyset. (inconsistent) • If a true statement results i.e. 0 = 0, then lines are same and solution set is the line itself. (dependent)
Practice Problem • Answer {(2,4)}
Practice Problem Hint: eliminate x first • Answer {(-7/2,-4)}
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 • 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.
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
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: • Answer = 16
Sample Cramer’s Rule problem • Solve by Cramer’s Rule
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.