Systems of Equations and Inequalities Chapter 4 Systems of Equations and Inequalities

Chapter Sections 4.1 – Solving Systems of Linear Equations in Two Variables 4.2 – Solving Systems of Linear Equations in Three Variables 4.3 – Systems of Linear Equations: Applications and Problem Solving 4.4 – Solving Systems of Equations Using Matrices 4.5 – Solving Systems of Equations Using Determinants and Cramer’s Rule 4.6 – Solving Systems of Linear Inequalities Chapter 1 Outline

Solving Systems of Linear Equations in Two Variables § 4.1 Solving Systems of Linear Equations in Two Variables

The solution to the above system is (1, 6). Definitions When two or more linear equations are considered simultaneously, the equations are called a system of linear equations. (1) y = x + 5 (2) y = 2x + 4 A solution to a system of equations in two variables is an ordered pair that satisfies each equation in the system. The solution to the above system is (1, 6).

Solutions Example: Determine if (–4, 16) is a solution to the system of equations. y = –4x y = –2x + 8 y = –2x + 8 16 = –2(–4) + 8 16 = 8 + 8 16 = 16 y = –4x 16 = –4(–4) 16 = 16 Yes, it is a solution

Solutions Example: Determine if (–2, 3) is a solution to the system of equations. x + 2y = 4 y = 3x + 3 x + 2y = 4 –2 + 2(3) = 4 –2 + 6 = 4 4 = 4 y = 3x + 3 3 = 3(–2) + 3 3 = –6 + 3 3 = –3 But… So it is NOT a solution

Types of Systems y = –4x y = –2x + 8 The solution to a system of equations is the ordered pair (or pairs) common to all lines in the system when the system is graphed. y = –4x y = –2x + 8 (–4, 16) is the solution to the system.

Types of Systems If the lines intersect in exactly one point, the system has exactly one solution and is called a consistent system of equations.

Types of Systems If the lines are parallel and do not intersect, the system has no solution and is called an inconsistent system.

Types of Systems If the two equations are actually the same and graph the same line, the system has an infinite number of solutions and is called a dependent system.

Solving Graphically To Obtain a Solution to a System of Equations Graphically Graph each equation and determine the point or points of intersection. Example: Solve the following system of equations graphically. Let x = 0; then y = 2. (0, 2) y = x + 2 y = -x + 4 Let y = 0; then x = -2. (-2, 0) Let x = 0; then y = 4. (0, 4) Let y = 0; then x =4. (4, 0) 1. Find the x- and y-intercepts. 2. Draw the graphs.

Solving Graphically Graph both equations on the same axes. The solutions is the point of intersection of the two lines, (1, 3).

Solve by Substitution The substitution method for solving a system of equations can be used to find the solution to a system. The goal is to obtain one equation containing only one variable.

Solve by Substitution To Solve a Linear System of Equations by Substitution Solve for a variable in either equation. If possible, solve for a variable with a numerical coefficient of 1 to avoid working with fractions. Substitute the expression found for the variable in step 1 into the other equation. This will result in an equation containing only one variable. Solve the equation obtained in step 2. Substitute the value found in step 3 into the equation from step 1. Solve the equation to find the remaining variable. Check your solution in all equations in the system.

Solve by Substitution Example: Solve the following system of equations by substitution. y = 3x – 5 y = -4x + 9

Solve by Substitution Since both equations are already solved for y, we can substitute 3x – 5 for y in the second equation and then solve for the remaining variable, x.

Solve by Substitution Now find y by substituting x = 2 into the first equation. Thus, we have x = 2 and y = 1, or the ordered pair (2, 1). A check will show that the solution to the system of equation is (2, 1).

Addition Method The addition (or elimination) method for solving a system of equations can also be used to find the solution to a system. This method is generally the easiest one to use. Again, the goal is to obtain one equation containing only one variable. Example: Solve by using the addition method. 2x + 5y = 3 3x – 5y = 17

Addition Method 2x + 5y = 3 3x – 5y = 17 Adding the equations yields: The y-variables are eliminated. x = 4 The x-variable can now be obtained using the same steps used for the substitution method.

Addition Method 2x + 5y = 3 3x – 5y = 17 Substitute x = 2 into either equation: 2x + 5y = 3 2(4) + 5y= 3 8 + 5y = 3 5y = -5 y = -1 The solution to this system is (4, -1). Don’t forget to check your answer!

Addition Method Steps To Solve a System of Equations by the Addition Method If necessary, rewrite each equation in standard form, ax + by = c. If necessary, multiply one or both equations by a constant(s) so that when the equations are added, the sum will contain only one variable. Add the respective sides of the equations. This will result in a single equation containing only one variable. Solve the equation obtained in step 3. Substitute the value found in step 4 into either of the original equations. Solve that equation to find the value of the remaining variable. Write the solution as an ordered pair. Check your solution in all equations in the system.

Addition Method Example: Solve by using the addition method. 2x + y = 11 x + 3y = 18 Since adding the equations at this point would not eliminate a variable, the first equation is multiplied by –2. – 2(x + 3y = 18) – 2x - 6y = –36 Now the equations are added. + 2x + y = 11 y = 5

Addition Method Example continued. Now solve for x by substituting 5 for y in either of the original equations. The solution is (3, 5).

Addition Method Example: Solve by using the addition method. x – 3y = 4 -2x + 6y = 1 Rewrite the equations so all the variables are on the left. Multiply equation 1 by 2. Set up the equations to add. – 2(x - 3y = 4) 2x – 6y = 8 -2x + 6y = 1 + 0 = 9 Everything on the left cancels. Since 0 =9 is a false statement, this system has no solution. The system is inconsistent and the graphs of these equations are parallel lines.