Solving Two-Variable Systems of Linear Equations Digital Lesson Solving Two-Variable Systems of Linear Equations

System of Linear Equations A set of linear equations in two variables is called a system of linear equations. 3x + 2y = 14 2x + 5y = 3 A solution of such a system is an ordered pair which is a solution of each equation in the system. Example: The ordered pair (4, 1) is a solution of the system since 3(4) + 2(1) = 14 and 2(4) – 5(1) = 3. Example: The ordered pair (0, 7) is not a solution of the system since 3(0) + 2(7) = 14 but 2(0) – 5(7) = – 35, not 3. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. System of Linear Equations

Solutions of Linear Equations Systems of linear equations in two variables have either no solutions, one solution, or infinitely many solutions. y x unique solution y x infinitely many solutions y x no solutions A system of equations with at least one solution is consistent. A system with no solutions is inconsistent. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Solutions of Linear Equations

Example: Solve the System x – y = –1 To solve the system by the graphing method, graph both equations and determine where the graphs intersect. 2x + y = 4 y x (1, 2) x – y = –1 2x + y = 4 The ordered pair (1, 2) is the unique solution. The system is consistent since it has solutions. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Example: Solve the System

Inconsistent Solutions Example: Solve the system by the graphing method. x – 2y = – 4 3x – 6y = 6 y x x – 2y = – 4 3x – 6y = 6 The lines are parallel and have no point of intersection. The system has no solutions and is inconsistent. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Inconsistent Solutions

Infinitely Many Solutions x – 2y = – 4 3x – 6y = – 12 Example: Solve the system by the graphing method. y x 3x – 6y = – 12 x – 2y = – 4 The graphs of the two equations are the same line and the intersection points are all the points on this line. The system has infinitely many solutions. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Infinitely Many Solutions

Definition: Algebraic Methods To solve a system by the substitution method: 1. Select an equation and solve for one variable in terms of the other. 2. Substitute the expression resulting from Step 1 into the other equation to produce an equation in one variable. 3. Solve the equation produced in Step 2. 4. Substitute the value for the variable obtained in Step 3 into the expression obtained in Step 2. 5. Check the solution. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Definition: Algebraic Methods

Example: Substitution Method 2x + y = 8 x – 3y = – 3 Example: Solve the system by the substitution method. 1. From the second equation obtain x = 3y – 3. 2. Substitute this expression for x into the first equation. 2(3y –3) + y = 8 3. Solve for y to obtain y = 2. 4. Substitute 2 for y in x = 3y – 3 and conclude x = 3. The solution is (3, 2). 5. Check: 2(3) – (2) = 8 (3) – 3(2) = –3 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Example: Substitution Method

Example: Substitution Method 2x – y = 10 4x – 2y = 8 Example: Solve the system by the substitution method. 1. From the first equation obtain y = 2x – 10. 2. Substitute 2x – 10 for y into the second equation to produce 4x – 2(2x – 10) = 8. 3. Attempt to solve for x. 4x – 2(2x – 10) = 8 4x – 4x + 20 = 8 20 = 8 False statement Because there are no values of x and y for which 20 equals 8, this system has no solutions. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Example: Substitution Method

Definition: Addition Method To solve a system by the addition (or elimination) method: 1. Multiply either or both equations by nonzero constants to obtain opposite coefficients for one of the variables in the system. 2. Add the equations to produce an equation in one variable. Solve this equation. 3. Substitute the value of the variable found in Step 2 into either of the original equations to obtain another equation in one variable. Solve this equation. 4. Check the solution. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Definition: Addition Method

Example: Addition Method Example: Solve the system by the addition method. 5x + 2y = 11 3x + 4y = 15 1. Multiply the first equation by –2 to make the coefficients of y opposites. – 10 x – 4y = – 22 3x + 4y = 15 2. Add the equations to obtain – 7x = –7. Therefore x = 1. 3. Substitute 1 for x in the first equation to produce 5(1) + 2y = 11 2y = 6 Therefore y = 3. The solution is (1, 3). 4. Check: 5(1) + 2(3) = 11 3(1) + 4(3) = 15 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Example: Addition Method

Example: Application Problem Example: A plane with a tailwind flew 1920 mi in 8 hours. On the return trip, against the wind, the plane flew the same distance in 12 hours. What is the speed of the plane in calm air and the speed of the tailwind? Let x = speed of the plane in calm air y = speed of the tailwind Use the formula Rate × Time = Distance. 12(x - y) 12 x - y Against Wind 8(x + y) 8 x + y With Wind Distance Time Rate This yields a system of equations in x and y. 8x + 8y = 1920 12x – 12y = 1920 Use the addition method to solve the system. Example continues Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Example: Application Problem

Example continued: Solve the system using the addition method. 8x + 8y = 1920 12x – 12y = 1920 Example continued: Solve the system using the addition method. 1. Multiply the first equation by 3 and the second equation by 2. 3(8x + 8y) = 3(1920) 2(12x – 12y) = 2(1920) 24x + 24y = 5760 24x – 24y = 3840 2. Add the equations to obtain 48 x = 9600. Therefore x = 200. 3. Substitute 200 for x in the first equation. 8(200) + 8y = 1920 y = 40 4. In 8 hours with an airspeed of 240 mph the plane will travel 8 × 240 = 1920 mi on the first leg of its flight. In 12 hours with an airspeed of 160 mph the plane will travel 12 × 160 = 1920 on the return leg of its flight. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Example continued