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Mrs. Branson Algebra 1 Chapter 6: Systems of Linear Equations and Inequalities Chapter Objectives: Lesson 6.1: Graphing Systems of Equations Determine the number of solutions a system of linear equations has, if any HW: Pg. 336 # 10–20 Evens and #26, 30 Lesson 6.2: Substitution Solve systems of equations by using substitution HW: Pg. 345 – 346 #10–26 Evens Lesson 6.3: Elimination Using Addition and Subtraction Lesson 6.4: Elimination Using Multiplication Solve systems of equations by using elimination with addition, subtraction, and multiplication HW: Pg. 352 #14-30 Evens and Pg. 358 #14-24 Evens Lesson 6.5: Applying Systems of Linear Equations Determine the best method for solving systems of equations Apply systems of equations HW: Pg. 365-366 #6-22 Evens Lesson 6.6: Organizing Data Using Matrices Organize data in matrices Perform matrix operations HW: Worksheet Lesson 6.7: Using Matrices to Solve Systems of Equations Write systems of equations as augmented matrices Solve systems of equations by using elementary row operations (row reduction) HW: Practice Test Part 1 Standards: A.REI.6: Solve systems of linear equations exactly and approximately (e.g. with graphs), focusing on pairs of linear equations in two variables. A.REI.7: Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between y = -3x and the circle x2 + y2 = 3. A.REI.5: Prove that given a system of two equations in two variables replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions. A.REI.8: Represent a system of linear equations as a single matrix equation in a vector variable. A.REI.9: Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 x 3 or better) Lesson 6.1: Graphing Systems of Equations Determine the number of solutions a system of linear equations has, if any HW: Pg. 336 # 10–20 Evens and #26, 30 Term Meaning Two or more equations with the same variables Example: A system with at least one solution Graphs intersect at one point or are the same line A consistent system with exactly one solution A consistent system with unlimited solutions that satisfy both equations A system with no solution The graphs are parallel Possible Solutions Number of Solutions Terminology Graph Examples: Consistent and Independent Consistent and Dependent Inconsistent Lesson 6.2: Substitution Solve systems of equations by using substitution HW: Pg. 345 – 346 #10–26 Evens An algebraic method to find an exact solution to a system of equations Steps for Solving a System of Equations using Substitution 1. When necessary, ________________ at least one equation for ___________________________ 2. ____________________ the resulting expression from Step 1 into the other equation to _________________ the variable. Then _____________________________________________ 3. ____________________ the value from Step 2 into ____________________________________ and solve for the other variable. 4. Write the ____________________ as an _____________________________________________ Examples: 𝑦 = 5𝑥 + 1 4𝑥 + 𝑦 = 10 𝑦 = 3𝑥 − 2 𝑦 = 2𝑥 − 5 −1 = 2𝑥 − 𝑦 8𝑥 − 4𝑦 = −4 3𝑥 + 𝑦 = −5 6𝑥 + 2𝑦 = 10 In 2000, the demand for nurses was 2,000,000, while the supply was only 1,890,000. The projected demand for nurses in 2010 was 2,820,000,000, while the supply was only projected to be 1,810,000. a. Define the variables, and write equations to represent these situations b. Use substitution to determine during which year the supply of nurses was equal to the demand.