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Sullivan Algebra and Trigonometry: Section 11.1 Systems of Linear Equations

Sullivan Algebra and Trigonometry: Section 11.1 Systems of Linear Equations. Objectives of this Section Solve Systems of Equations by Substitution Solve Systems of Equations by Elimination Identify Inconsistent Systems Express the Solutions of a System of Dependent Equations.

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Sullivan Algebra and Trigonometry: Section 11.1 Systems of Linear Equations

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  1. Sullivan Algebra and Trigonometry: Section 11.1Systems of Linear Equations • Objectives of this Section • Solve Systems of Equations by Substitution • Solve Systems of Equations by Elimination • Identify Inconsistent Systems • Express the Solutions of a System of Dependent Equations

  2. A system of equations is a collection of two or more equations, each containing one or more variables. To solve a system of equations means to find all solutions of the system. When a system of equations has at least one solution, it is said to be consistent; otherwise it is called inconsistent.

  3. If the graph of the lines in a system of two linear equations in two variables intersect, then the system of equations has one solution, given by the point of intersection. The system is consistent and the equations are independent. y Solution x

  4. If the graph of the lines in a system of two linear equations in two variables are parallel, then the system of equations has no solution, because the lines never intersect. The system is inconsistent. y x

  5. If the graph of the lines in a system of two linear equations in two variables are coincident, then the system of equations has infinitely many solutions, represented by the totality of points on the line. The system is consistent and dependent. y x

  6. Two Algebraic Methods for Solving a System 1. Method of substitution2. Method of elimination STEP 1:STEP 2: Solve for x in (2) Substitute into (1)

  7. STEP 3: Solve for y STEP 4: Substitute y = –4 into (2) Solution: (3, –4)

  8. STEP 5: Verify solution Solution: (3, –4) (1): (2):

  9. Multiply (1) by 3 When adding these 2 equations, you get: 0x + 0y = 33 This equation has no solution so the system is inconsistent. The lines are parallel.

  10. Consistent & Dependent.

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