9.1 - Systems of Linear Equations: Two Variables
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1 9.1 - Systems of Linear Equations: Two Variables Recall that a system of equations consists of two or more equations each with two or more variables. A solution to a system in two variables is an ordered pair (x, y) that satisfies each equation in the system simultaneously. Likewise, a solution to a system in three variables is an ordered triple (x, y, z), and so on. For now, we will concentrate on systems of linear equations. A system of two linear equations in two variables can have one solution, no solution, or an infinite number of solutions. Consistent System Intersecting lines - One solution Inconsistent System Parallel lines - No solution Dependent System Same line - Infinite solutions Methods of Solution Recall that our two algebraic methods of solution are substitution and elimination. Substitution: Solve for one of the variables in one of the equations and substitute it into the other equation. Elimination: Add a multiple of one equation to the other in order to eliminate a variable. Example 1 Solve x 3y = 2 5x + 3y = 17 using the substitution method. Chapter 9 - Systems of Equations and Inequalities Page 1 of 10
2 Example 2 Solve 3x 2y = 6 x + 4y = 4 using the method of elimination. Example 3 Solve 4x + 2y = 3 10x + 4y = 1 using the method of elimination. Example 4 So what happens algebraically when there is no solution or when there are an infinite number of solutions? 2x 5y = 7 (a) 4x + 10y = 2 (b) 2x 5y = 7 4x + 10y = 14 Chapter 9 - Systems of Equations and Inequalities Page 2 of 10
3 9.2 - Systems of Linear Equations: Three Variables In this section, we are going to solve higher-order systems; i.e., systems with more than two variables and two equations. Our approach will be to apply elimination to reduce the system. Example 1 x + 2z = 5 Solve y 30z = 16. x 2y + 4z = 8 Here is what can happen graphically with a system of three equations in three variables: Chapter 9 - Systems of Equations and Inequalities Page 3 of 10
4 Example 2 x + y z = 1 Solve 3x y + 2z = 9. 5x + 3y + 3z = 1 Example 3 3x 2y + z = 4 Solve 3z = 9. Chapter 9 - Systems of Equations and Inequalities Page 4 of 10
5 9.6 - Solving Systems with Gaussian Elimination Matrix and Dimension A matrix is a rectangular array of numbers, which are called entries. The dimension (or size) of a matrix is referred to by the number of rows by the number of columns; i.e., rows columns. Example 1 Give the dimensions of each matrix ( ) (a) (b) (c) ( ) 1 (d) 2 3 We can use matrices to solve systems of equations. Each row will represent an equation and each column will keep track of the variables. x + y z = 1 Consider the system 3x y + 2z = 9. 5x + 3y + 3z = 1 Example The coefficient matrix for the system is given by The augmented matrix for this system is given by Write the augmented matrix in system form and solve it. Chapter 9 - Systems of Equations and Inequalities Page 5 of 10
6 Note how this stair-case pattern made it easy to solve the system. We call this upper-triangular form and we solved it using what is called back-substitution. The method of Gaussian elimination will help us get systems into this form. We will use 3 operations that produce equivalent systems (systems with the same solution set). The elementary row operations used to transform a system into an equivalent system are as follows: Interchange two rows, denoted by R i R j. Multiply a row by a nonzero constant, denoted by cr i R i. Add a multiple of one row to another row to replace the latter row, denoted by cr i + R j R j. The form that we will obtain is called row-echelon form. This will be the stair-case pattern seen above, with each leading coefficient being 1. The two forms we will work toward when doing elimination are called row-echelon form and reduced row-echelon form. Each form can be obtained using the three elementary operations. Row-Echelon Form Gaussian Elimination Reduced Row-Echelon Form a b c Gauss-Jordan Elimination Example 2 x + y z = 1 Solve 3x y + 2z = 9 5x + 3y + 3z = 1 using Gaussian elimination. Chapter 9 - Systems of Equations and Inequalities Page 6 of 10
7 Example 3 x + 4y z = 4 Solve 2x + 5y + 8z = 15 x + 3y 3z = 1 using Gaussian elimination. Chapter 9 - Systems of Equations and Inequalities Page 7 of 10
8 We can use our graphing calculators to solve systems (instructions for TI-83/84): Select MATRIX (2nd - x 1 ). Move to the column labeled EDIT and select a matrix on the list. Enter the size of the matrix and press enter. Enter the numbers for your augmented matrix, making sure to press ENTER after each one (especially the last one). Now QUIT (2nd-MODE) this screen. Select MATRIX again and move to the column labeled MATH. Scroll down and you will find options for REF and RREF, which correspond to row-echelon form and reduced row-echelon form, respectively. If we are going to use technology, there is no reason to go half of the way, so select RREF. Select MATRIX again and select your matrix in the NAMES column. Press ENTER. Example 4 x 2y + 3z = 9 Solve x + 3y = 4 2x 5y + 5z = 17 using Gaussian elimination and check your answer using your graphing calculator. Example 5 2a + b + c + d = 1 Solve a + 3b 3c 3d = 0 3a 4b + 2c + 2d = 1 using your graphing calculator. Chapter 9 - Systems of Equations and Inequalities Page 8 of 10
9 9.3 - Systems of Nonlinear Equations and Inequalities: Two Variables So far we have found intersections involving linear functions. What if we wanted to determine the flight path needed for a space shuttle to dock with a station in orbit around the earth? This might be the intersection of part of a parabola with a circle. So how could we find these intersections? We use the same methods that we use for linear systems, though substitution will prove more useful than elimination most of the time. Example 1 3x y = 2 Solve the system 2x 2. Include a sketch showing the curves and their intersections. y = 0 Example 2 x 2 + y 2 = 10 Solve the system. Include a sketch showing the curves and their intersections. x 3y = 10 Chapter 9 - Systems of Equations and Inequalities Page 9 of 10
10 Example 3 Solve the system 4x 2 + y 2 = 13 x 2 + y 2 = 10. Example 4 Find the intersections of the hyperbola and ellipse shown in the graph. Chapter 9 - Systems of Equations and Inequalities Page 10 of 10
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