Chapter 4  Systems of Equations and Inequalities


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1 Math Spring 2009 Chapter 4  Systems of Equations and Inequalities 4.1 Solving Systems of equations in Two Variables Definition 1. A system of linear equations is two or more linear equations to which we try to find a common solution. EX 1. Consider the equations: y = x + 4 y = 2x + 1 A solution to a system of linear equtions is an ordered pair or pairs that satisfy ALL equations in the system. Let s verify that (3, 7) is a solution to the given system Solve Systems of Linear Equations Graphically To solve a system with two equations in two variables we graph them both on the same axes and common solution is the point where they intersect. EX Solve the following system of equations graphically. y = x + 3 y = x Let s consider all the possible ways that two lines can interact: (a) (b) (c) 1
2 4.1.2 Solving Systems by Substitution It is usually not practical to solve systems by graphing (messy drawings, points may not be exactly clear) so we need more precies methods which we now develop. Our first algebraic method is called substitution. Substitution Method 1. Pick an equation and solve for one of the variables. (Note: It is usually easiest to solve for a variable with coefficient of 1.) 2. Substitute the expression found for the variable in step 1 into the other equation. We now have an equation with only one variable. 3. Solve the equation in step 2 to find the value of one of the variables. 4. Substitute the value found in step 3 into the equation from step 1. Solve the equation to find the other variable. 5. Check your solution in all of the equations in the system. EX Solve using the substitution method. y = 2x 10 y = 3x Solve using the substitution method. 3x + y = 1 x + 2y = Solving Systems Using the Addition Method Often the easiest method for solving a system of equations is the addition method (also known as the elimination method). Before we list the steps, let s look at an example. 2
3 EX 4. Solve the following system of equations using the addition method. x + 4y = 9 3x 4y = 11 Addition Method 1. If necessary, rewrite each equation in standard form. 2. If necessary, multiply one or both equations by a constant so that when the equations are added, the sum will contain only one variable. 3. Add the equations. We get an equation in one variable. 4. Solve the equation found in step Substitute the value found in step 4 into either of the original equations and solve for the other variable. 6. Check your solution in all of the equations in the system. EX Solve using the addition method: 3x + y = 1 x + 2y = Solve using the addition method: 3x + 2y = 5 2x + 3y = 4 3
4 3. Solve using the addition method: x = 13 5y 1 2 x = 1 4. Solve using the addition method: x + 4y = 7 3x 12y = Solving Systems of Linear Equations in Three Variables An equation of the form 3x 2y + 4z = 11 is a linear equation in three variables. The solution to such an equation is an ordered triple of the form (x, y, z). For, example a solution of the above equation is (1, 2, 3). We will consider systems of three equations in three variables. To solve systems of equations in three variables we will use the methods we have already introduced, namely the substitution and addition methods. EX Solve the following system using the substitution method. 2x = 8 2x + y = 3 3x + 5y + 6z = 1 4
5 2. Solve the following system using the addition method. 2x + 3y + 4z = 4 x  2z = 12 y  3z = Solve the following system using the addition method. 2x + 3y + z = 4 7x + 4y + 2z = 10 6x  9y  3z = Applications and Problem Solving There are countless applications for systems of equations. We will examine a couple examples. EX How many pints of a 10% salt solution and a 50% salt solution must be combined to get 44 pints of a 40% salt solution? 5
6 2. On an algebra test, a total of 75 students had grades of A or B. There were 5 more students with A s than B s. Find the number of students who had an A and the number of students who had a B. 3. Three brothers invest a total of $35,000 a business. The first brother earns 3% profit on his investment. The second brother earns a 5% profit on his investment. The third brother earns 7% profit on his investment. The third brother invested $5000 more than the sum of the other two brothers. After one year, the profit earned was $2050. How much did each brother invest? 4.4 Solving Systems of Equations Using Matrices Matrices A matrix is a rectangular array of numbers within brackets. The plural of matrix is matrices. EX 8. Here are a couple matrices: [ ] or [ The numbers inside the matrices are referred to as elements. We refer to the size of the matrix by the number of rows and columns, thus the left matrix has 2 rows and 3 columns so it is a 2 3 matrix. A square matrix is a matrix which has the same number of rows as columns. We will use matrices to solve our systems of equations. In order to do this we will represent our system with an augmented matrix. Consider the system of equations a 1 x + b 1 y = c 1 a 2 x + b 2 y = c 2 the corresponding matrix is written: [ ] a1 b 1 c 1 c 2 ] 6
7 EX 9. Express the following system of linear equations with an augmented matrix. 3x 2y = 9 12x + 5y = Solve Systems of Linear Equations In order to solve systems of equations using matrices we will attempt to transform our matrices using 3 simple transformations. Our goal is to rewrite the matrix in the following form (called triangular form): [ ] 1 a p 0 1 q To transform a matrix into this form we have the following row transformations: Procedures for Row Transformations 1. Any row may be multiplied (or divided) by a nonzero number. 2. All the numbers in a row may be multiplied by any nonzero real number, then added to any other row. 3. The order of the rows may be switched. EX Solve the following system of equations using matrices. 3x 2y = 9 12x + 5y = 10 7
8 2. Solve the following system of equations using matrices. 4x + 3y = 2 x + 3y = 4 3. Solve the following system of equations using matrices. 2x 3y = 3 4x + 9y = Recognize Inconsistent and Dependent Systems Recall that we call a system of equations inconsistent if there are no common solution to the equations of the system (parallel lines). A dependent system is one with an infinite number of solutions. Let s examine how to determine, using matrices, if a system is inconsistent or dependent. EX Solve the system of equations using matrices. 2x 4y = 7 3x + 6y = 8 8
9 2. Solve the system of equations using matrices. x 3y = 4 2x 6y = Determinants and Cramer s Rule Evaluate a Determinant of a 2 2 Matrix For every 2 2 matrix we can find a number [ associated ] with it called the determinant. a1 b The determinant of a 2 2 matrix 1 is denoted a 1 b 1 and is evaluated as follows: a 1 b 1 = a 1b 2 a 2 b 1 EX 12. Evaluate the following determinants: Cramer s Rule We will study now a cool rule that gives us a formula for solving systems of linear equations. We will only study this for systems with two variables but it works for three or more variable system of equations. First some notation. Given the system of equations: a 1 x + b 1 y = c 1 a 2 x + b 2 y = c 2 We denote by D the following: D = a 1 b 1 = a 1b 2 a 2 b 1 Further we denote with D x and D y : D x = c 1 b 1 c 2 b 2 = c 1b 2 c 2 b 1 9
10 and D y = a 1 c 1 a 2 c 2 = a 1c 2 a 2 c 1 Finally, we have Cramer s Rule: For the system of equations Then the solution is given by and a 1 x + b 1 y = c 1 a 2 x + b 2 y = c 2 x = D x D = c 1 b 1 c 2 b 2 a 1 b 1 y = D y D = a 1 c 1 a 2 c 2 a 1 b 1 EX 13. Solve the system of equations using Cramer s rule. 3x + 5y = 9 4x y = 11 REMARK 1. When D = 0: If D = 0, D x = 0, and D y = 0, then the system is dependent. If D = 0 and either D x 0 or D y 0, then the system is inconsistent. 4.6 Solving Systems of Linear Inequalities Recall, we learned how to graph linear inequalities in two variables: EX 14. Graph the inequality 2x + 3y > 6 We now investigate how to solve systems of linear inequalities graphically. To do this, we graph each inequality on the same axes. The solution is the set of points whose coordinates satisfy all the inequalities in the system. 10
11 EX Graph: y 1 3 x + 6 x y < 2 2. Graph 2x y 4 3x + 2y > 6 3. Graph x 0 y 0 x + y 5 3x + y 9 11
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