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Chapter 8. Systems of Equations and Inequalities. 8.1 Systems of Equations. A system of equations is a set of equations with common variables. A solution is an ordered pair that is true in all of the equations. Doesn’t matter to me which one you choose to use. Substitution.
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Chapter 8 Systems of Equations and Inequalities
8.1 Systems of Equations • A system of equations is a set of equations with common variables. • A solution is an ordered pair that is true in all of the equations. • Doesn’t matter to me which one you choose to use.
Substitution • Solve for one variable in one of the equations. • Substitute this expression into the other equation to get one equation with one unknown. • Back substitute the value found in step 2 into the expression from step 1. • Check.
Solve: Step 1: Solve equation 1 for y. Step 2: Substitute this expression into eqn 2 for y. Step 3: Solve for x. Step 4: Substitute this value for x into eqn 1 and find the corresponding y value. Step 5: Check in both equations.
Elimination • Adjust the coefficients. • Add the equations to eliminate one of the variable. Then solve for the remaining variable. • Back substitute the value found in step 2 into one of the original equations to solve for the other variable. • Check.
Solve: Step1: The coefficients on the y variables are opposites. No multiplication is needed. Step 2: Add the two equations together. Step 3: Solve for x. Step 4: Substitute this value for x into either eqn and find the corresponding y value. Step 5: Check in both equations.
8.2 Systems of Linear Equations in Two Variables • When solving a system, we are looking for where two lines intersect. • Two methods: Elimination and substitution. • Be comfortable with both methods.
Three Possible Scenarios Scenario 1: Two lines intersect in one point. The system is “independent and consistent.”
Three Possible Scenarios Scenario 2: Two lines don’t intersect. They are parallel. The system is “independent and inconsistent.”
Three Possible Scenarios Scenario 3: Identical lines which intersect at an infinite number of points. The system is “dependent and consistent.”
8.3 Matrices and Systems of Equations We can translate a given system of equations into an augmented matrix. With two rows and two columns, this matrix is a 2x3 matrix.
Gaussian Elimination To solve a system of equations using Gaussian elimination with matrices, we use the same rules as before. • Interchange any two rows. • Multiply each entry in a row by the same nonzero constant. • Add a nonzero multiple of one row to another row.
Example 1 Solve: Matrix:
Chapter 8 Section 3 Example • Solve the system of equations using Gauss-Jordan Method
Chapter 8 Section 3 Example • Solve the system of equations using Gauss-Jordan Method
Chapter 8 Section 3 Example • Solve the system of equations using Gauss-Jordan Method
Chapter 8 Section 3 Example • Solve the system of equations using Gauss-Jordan Method
Chapter 8 Section 3 Example • Solve the system of equations using Gauss-Jordan Method
Chapter 8 Section 3 Example • Solve the system of equations using Gauss-Jordan Method
Chapter 8 Section 3 Example • Solve the system of equations using Gauss-Jordan Method
Chapter 8 Section 3 Example • Solve the system of equations using Gauss-Jordan Method
Chapter 8 Section 3 Example • Solve the system of equations using Gauss-Jordan Method
Chapter 8 Section 3 Example • Solve the system of equations using Gauss-Jordan Method
Chapter 8 Section 3 Example • Solve the system of equations using Gauss-Jordan Method
Chapter 8 Section 3 Example • Solve the system of equations using Gauss-Jordan Method
Chapter 8 Section 3 Example • Solve the system of equations using Gauss-Jordan Method
Chapter 8 Section 3 Example • Solve the system of equations using Gauss-Jordan Method
Chapter 8 Section 3 Example • Solve the system of equations using Gauss-Jordan Method (0, 2, 1)
8.4 Systems of Equations: Matrices Definition: An m X nmatrix is a rectangular array of numbers with mrows and ncolumns. The numbers are the entries of the matrix. The subscript on the entry indicates that it is in the ith row and the jth column
Augmented Matrix Linear System Augmented Matrix
Elementary Row Operations • Add a multiple of one row to another. • Multiply a row by a nonzero constant. • Interchange two rows. Symbol Description Change the ith row by adding k times row j to row i, putting the result back in row i. Multiply the ith row by k. Interchange row i and row j.
Example Solve: Matrix:
Row-Echelon Form and Reduced Row-Echelon Form A matrix is in row-echelon form if it satisfies the following conditions. • The first nonzero entry in each row (left to right) is 1. This is called a leading 1. • The leading entry in each row is to the right of the leading entry in the row immediately above it. • Every number above and below each leading entry is a one. This is called reduced row-echelon form.
Inconsistent and Dependent Systems A leading variable is a linear system is one that corresponds to a leading entry in the row-echelon form of the matrix of the system. Suppose the system has been transformed into row-echelon form. Then exactly one of the following is true. • No solution. There is a row that represents 0 = C, where C is not zero. The system has no solution and is inconsistent. • One solution. If each variable is a leading variable, then the system has exactly one solution. • Infinitely many solutions. If there is at least one row of all zeros, the system has infinitely many solutions. The system is called dependent.
