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Solving Systems of Equations

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Key Terms 0 Solution 0 A solution of a system of equations is an ordered pair that satisfies each equation in the system. 0 Solving the system of equations 0 Finding the set of all solutions is called solving the system of equations.

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The Method of Substitution 0 Steps 0 Solve one of the equations for one variable in terms of the other. 0 Substitute the expression found in step 1 into the other equation to obtain an equation in one variable. 0 Solve the equation in step 2.

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Steps Continued 0 Back-substitute the value(s) obtained in Step 3 into the expression obtained in Step 1 to find the value(s) of the other variable. 0 Check that each solution satisfies both of the original equations.

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Example Original Step 1: Solve equation 1 for x in terms of y. Equation 1 Equation 2 Step 2: Substitute y for x in equation 2 to obtain an equation in one variable.

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Example Continued Solve the equation obtained in Step 2. Back-substitute the value(s) obtained in Step 3 into the expression obtained in Step 1 to find the value(s) of the other variable.

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Practice

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How do you use substitution to solve systems of equations?

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Word Problem 0 Dorothy is 3 times as old as her sister. In 5 years she will be twice as old as her sister. How old are Dorothy and her sister now? In 5 years Dorothy will be D+5 and her sisters age will be S+5 Right now

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Word Problem Continued If she will be 2 times as old as her sister. Substitute from equation one D=3S

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The Method of Elimination 0 The key step in the method of elimination is to obtain, for one of the variables, coefficients that differ only in sign so that adding the equations eliminates the variable.

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The Method of Elimination 0 To use the method of elimination to solve a system of two linear equations in x and y, perform the following steps. 0 Obtain coefficients for x (or y) that differ only in sign by multiplying all terms of one or both equations by suitable chosen constants. 0 Add the equations to eliminate one variable; solve the resulting equation.

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The Method of Elimination Continued 0 Back-substitute the value obtained in Step 2 into either of the original equations and solve for the other variable. 0 Check your solution in both original equations.

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Example Equation 1 Equation 2 Add the equations.

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Example Continued Substitute 2 for y to solve for x Solution Check the Solution!!!

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Example Equation 1 Equation 2

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Example Continued Multiply equation 1 by 4. Multiply Equation 2 by 3. Add equations.

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Example Continued Add Equations Solve for x.

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Example Continued By back-substituting the value for x in equation 2, you can solve for y. Solution

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Graphical Interpretation of Solutions 0 For a system of two linear equations in two variables, the number of solutions is one of the following. Number of SolutionsGraphical Interpretation 1. Exactly one solution.The two lines intersect at one point. 2. Infinitely many solutions.The two lines are coincident (identical). 3. No SolutionThe two lines are parallel.

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Consistent or Inconsistent 0 A system of linear equations is consistent if it has at least one solution. It is inconsistent if it has no solution.

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Inconsistent System

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Consistent System

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Quick Quiz 1. The first step in solving a system of equations by the method of is to obtain coefficients for x (or y) that differ only in sign. 2. Two systems of equations that have the same solution set are called systems. 3. A system of linear equations that has at least one solution is called, whereas a system of linear equations that has no solution is called.

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