Algebra 2 Notes Systems of Equations and Inequalities Unit 03c. System of Equations in Three Variables

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1 System of Equations in Three Variables Big Idea A system of equations in three variables consists of multiple planes graphed on the same coordinate plane. The solutions to these systems consists of a single point (ordered triple), a line (infinite number of solutions), a plane (infinite number of solutions), or no solution (planes do not intersect). Like two variable systems, systems of equations in three variables can be used to solve many real-world problems. Objectives: A.CED.3 A. Create equations that describe numbers or relationships - Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. Skills Solve a system of equations in three variables Solve real-world problems using systems of linear equations in three variables Vocabulary Ordered Triple - The solution of a system of equations in three variables x, y, and z. Algebra 2 Unit 03c Systems of Equaations in Three Variablesrev Page 1 of 6 6/20/2013

2 Examples, Notes, and Exam Questions Solving a System of Three Linear Equations (using substitution) Ex 1: Solve the system. 5x 2y z 7 x 2y 2z 0 3yz 17 Step One: Solve one of the equations for one of the variables. (Solve the third equation for z.) 3yz 17 z 17 3y Step Two: Substitute the expression from Step One into both of the other equations. 5x 2y 17 3y 7 5x 2y 17 3y 7 5x5y 10 x 2y y 0 x 2y 34 6y 0 x 8y 34 Step Three: Solve the system of two equations and two variables created in Step Two using substitution (or linear combinations). 5x5y 10 x 8y 34 x 8y34 5 8y 34 5y 10 40y170 5y 10 45y 180 y 4 x 8y34 x x 2 Step Four: Find the third variable by substituting the values from Step Three into the equation from Step One. z 17 3y z Step Five: Write your answer as an ordered triple, and check it in all three original equations. Solution: 2,4,5 Solving a System of Three Linear Equations (using linear combinations) x 5y z 16 Ex 2: Solve the system. 3x 3y 2z 12 2x 4 y z 20 Step One: Choose two of the equations and eliminate one of the variables using linear combinations. x 5y z x 15y 3z 48 3x 3y 2z x 3y 2z 12 18y5z 36 Algebra 2 Unit 03c Systems of Equaations in Three Variablesrev Page 2 of 6 6/20/2013

3 Step Two: Choose one of the equations from Step One and the leftover equation to eliminate the same variable using linear combinations. x 5y z x 10y 2z 32 2x 4y z x 4y z 20 6y3z 12 Step Three: Use the resulting equations from Steps One and Two. Solve for the remaining variables in this system of two equations using linear combinations. 18y 5z y 5z 36 6y 3z y 9z 36 4z 0 z 0 6y 3z 12 6y 12 6y y 2 Step Four: Find the third variable by substituting the values from Step Three into any of the three original equations. x 5y z 16 x x x 6 Step Five: Write your answer as an ordered triple, and check it in all three original equations. Solution: 6,2,0 Algebra 2 Unit 03c Systems of Equaations in Three Variablesrev Page 3 of 6 6/20/2013

4 Learn to solve three variable systems Solving a linear system of three variables is very similar to the process for solving a two variable system. Ex 3: Solve the following two variable system using the Elimination Method. 4x 9y 10 8x12y 8 Note: The solution of the system is the ordered pair (x, y) that satisfies both equations in the system. A three variable system is made up of three linear equations of the form ax by cz d, whose graph is three dimensional and is graphed on a three dimensional coordinate system. The coordinate system is made up of a vertical Z-axis, horizontal y-axis, and an x-axis defining the depth from front to back. Solutions of a three variable system are defined as the intersection point of all three lines and are known as an ordered triple with the following formx, y, z. Ex 4: x 2y 3z 3 2x 5y 4z 13 5x 4y z 5 Determine if the ordered triple 2, 1,1 is a solution of the system above. The ordered triple must satisfy all three equations to be a solution. Steps for solving a three variable system Ex 5: x 3y z 11 2x y z 1 5x 2y 3z 21 1 st. Use linear combination to eliminate one variable from two of the equations, which will give you a two variable system. 2 nd. Solve the two variable system for both variables. 3 rd. Substitute both values found is step 2 back into one of the original equations to find the third value. 4 th. Check your answer in all three original equations to be sure it is a solution. 5 th. Write your solution as an ordered triple (x,y,z). Algebra 2 Unit 03c Systems of Equaations in Three Variablesrev Page 4 of 6 6/20/2013

5 Ex 6: Solve the system. x 2y z 3 2x 2y z 5 4x 4y 2z 6 Ex 7: : Solve the system. 2x 4y z 1 3x 3y z 2 5x y z 8 Algebra 2 Unit 03c Systems of Equaations in Three Variablesrev Page 5 of 6 6/20/2013

6 SAMPLE QUESTIONS 1. Decide whether the given ordered triple is a solution of the system. 5, 6,1 2x 2y z 3 5y z 31 x 3y 2z Use Linear combination to solve the system. 9x 5y z 11 6x 4y 2z 2 2x 2y 4z 4 3. What is the x-coordinate of the solution to the system of equations? 2x y z 5 x 3z 14 2x 3 y 2z 2 A. 1 B. 3 C. 4 D. 5 Sample SAT Question(s): Taken from College Board online practice problems. 3x 2y 2z 19 3x y z If the equations above are true, which of the following is the value of y z? (A) 5 (B) 4 (C) 0 (D) 4 (E) 5 5. The sum of r and p is equal to twice s, and p is 36 less than twice the sum of r and s. What is the value of r? Grid-In Algebra 2 Unit 03c Systems of Equaations in Three Variablesrev Page 6 of 6 6/20/2013