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{ Solving a System of Equations Linear and Linear Inequalities

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A set of two or more equations in two or more variables A set of two or more equations in two or more variables Linear system- variables in each equation are all to the power of one Linear system- variables in each equation are all to the power of one Inequality- the equal sign has been replaced with less than, less than or equal to, greater than, or greater than or equal to Inequality- the equal sign has been replaced with less than, less than or equal to, greater than, or greater than or equal to What is a system?

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A set of values that satisfy all the equations in the system. A set of values that satisfy all the equations in the system. Solution Set

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3 Methods 3 Methods Solving by substitution- solve one equation for a variable and plug into the other Solving by substitution- solve one equation for a variable and plug into the other Solving by elimination- adding or subtracting one equation from the other Solving by elimination- adding or subtracting one equation from the other Graphing- graphing both equations and looking for intersection points Graphing- graphing both equations and looking for intersection points Methods of Solving

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Three possibilities for the number of solutions in a two equation system with two different variables Three possibilities for the number of solutions in a two equation system with two different variables No solution No solution One solution One solution Infinitely many solutions Infinitely many solutions Linear Systems

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1. Solve one equation for x or y. 1. Solve one equation for x or y. 2. Substitute the expression for x or y into the other equation 2. Substitute the expression for x or y into the other equation 3. Solve for the remaining variable 3. Solve for the remaining variable 4. Substitute the value found in Step 3 into one of the original equations, and solve for the other variable 4. Substitute the value found in Step 3 into one of the original equations, and solve for the other variable 5. Verify the solution in each equation 5. Verify the solution in each equation Solving by Substitution

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1. Multiply one or both of the equations by a nonzero constant so that the coefficients of x or y are opposites of one another 1. Multiply one or both of the equations by a nonzero constant so that the coefficients of x or y are opposites of one another 2. Eliminate x or y by adding the equations, and solve for the remaining variable 2. Eliminate x or y by adding the equations, and solve for the remaining variable 3. Substitute the value found in step 2 into one of the original equations and solve for the other variable 3. Substitute the value found in step 2 into one of the original equations and solve for the other variable 4. Verify the solution in each equation 4. Verify the solution in each equation Solving by Elimination

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{ One Step Further Word Problems

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Read the problem Read the problem Define variables Define variables Write out the two equations first Write out the two equations first Solve using substitution, graphing, or elimination Solve using substitution, graphing, or elimination Linear System Word Problems

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A ball game is attended by 575 people and total ticket sales are $2575. If tickets cost $5 for adults and $3 for children, how many adults and how many children attended the game A ball game is attended by 575 people and total ticket sales are $2575. If tickets cost $5 for adults and $3 for children, how many adults and how many children attended the game

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A café sells two kinds of coffee in bulk. The Costa Rican sells for $4.50 per pound and the Kenyan sells for $7.00 per pound. The owner wishes to mix a blend that would sell for $5.00 per pound. How much of each type of coffee should be used in the blend? A café sells two kinds of coffee in bulk. The Costa Rican sells for $4.50 per pound and the Kenyan sells for $7.00 per pound. The owner wishes to mix a blend that would sell for $5.00 per pound. How much of each type of coffee should be used in the blend?

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A toy company makes dolls, as well as collector cases for each doll. To make x cases costs the company $5000 in fixed overhead, plus $7.50 per case. An outside supplier has offered to produce any desired volume of cases for $8.20 per case. A toy company makes dolls, as well as collector cases for each doll. To make x cases costs the company $5000 in fixed overhead, plus $7.50 per case. An outside supplier has offered to produce any desired volume of cases for $8.20 per case. Write an equation that expresses the companys cost to make x cases Write an equation that expresses the companys cost to make x cases Write an equation that expresses the cost of buying x cases from the outside supplier Write an equation that expresses the cost of buying x cases from the outside supplier When should the company make cases themselves, and when should they buy them from the outside supplier? When should the company make cases themselves, and when should they buy them from the outside supplier?

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